Isometry Lie algebras of indefinite homogeneous spaces of finite volume

Let g be a real finite‐dimensional Lie algebra equipped with a symmetric bilinear form ⟨·,·⟩ . We assume that ⟨·,·⟩ is nil‐invariant. This means that every nilpotent operator in the smallest algebraic Lie subalgebra of endomorphisms containing the adjoint representation of g is an infinitesimal isometry for ⟨·,·⟩ . Among these Lie algebras are the isometry Lie algebras of pseudo‐Riemannian manifolds of finite volume. We prove a strong invariance property for nil‐invariant symmetric bilinear forms, which states that the adjoint representations of the solvable radical and all simple subalgebras of non‐compact type of g act by infinitesimal isometries for ⟨·,·⟩ . Moreover, we study properties of the kernel of ⟨·,·⟩ and the totally isotropic ideals in g in relation to the index of ⟨·,·⟩ . Based on this, we derive a structure theorem and a classification for the isometry algebras of indefinite homogeneous spaces of finite volume with metric index at most 2. Examples show that the theory becomes significantly more complicated for index greater than 2. We apply our results to study simply connected pseudo‐Riemannian homogeneous spaces of finite volume.


Introduction and main results
Let g be a finite-dimensional Lie algebra equipped with a symmetric bilinear form ·, · . The pair is called a metric Lie algebra. Traditionally, the bilinear form ·, · is called invariant if the adjoint representation of g acts by skew linear maps. We will call ·, · nil-invariant, if every nilpotent operator in the smallest algebraic Lie subalgebra of endomorphisms containing the adjoint representation of g is a skew linear map. This nil-invariance condition appears to be significantly weaker than the requirement that ·, · is invariant. Recall that the dimension of a maximal totally isotropic subspace is called the index of a symmetric bilinear form, and that the form is called definite if its index is zero. Since definite bilinear forms do not admit nilpotent skew maps, the condition of nil-invariance is less restrictive and therefore more interesting for metric Lie algebras with bilinear forms of higher index.
In this paper, we mainly study finite-dimensional real Lie algebras g with a nil-invariant symmetric bilinear form. We will discuss the general properties of these metric Lie algebras, compare them with Lie algebras with invariant symmetric bilinear form and derive elements of a classification theory, which give a complete description for low index, in particular, in the situation of index less than 3.

Nil-invariant bilinear forms and isometry Lie algebras
The motivation for this article mainly stems from the theory of geometric transformation groups and automorphism groups of geometric structures.
Namely, consider a Lie group G acting by isometries on a pseudo-Riemannian manifold (M, g) of finite volume. Then at each point p ∈ M , the scalar product g p naturally induces a symmetric bilinear form ·, · p on the Lie algebra g of G. As we show in Section 2 of this paper, the bilinear form ·, · p is nil-invariant on g. Note that, in general, ·, · p will be degenerate, since the subalgebra h of g tangent to the stabilizer G p of p is contained in its kernel.
Isometry groups of Lorentzian metrics (where the scalar products g p are of index one) have been studied intensely. Results obtained by Adams and Stuck [1] in the compact situation and by Zeghib [15] amount to a classification of the isometry Lie algebras of Lorentzian manifolds of finite volume.
In these works, it is used prominently that, for Lorentzian finite volume manifolds, the scalar products ·, · p are invariant by the elements of the nilpotent radical of g, cf. [1, § 4]. The latter condition is closely related to nil-invariance, but it is also significantly less restrictive. The role played by the stronger nil-invariance condition seems to have gone unnoticed so far.
Aside from Lorentzian manifolds, the classification problem for isometry Lie algebras of finite volume geometric manifolds with metric g of arbitrary index appears to be much more difficult.
Some more specific results have been obtained in the context of homogeneous pseudo-Riemannian manifolds. Here, M can be described as a coset space G/H, and any associated metric Lie algebra (g, ·, · p ) locally determines G and H, as well as the geometry of M . These pseudo-Riemannian manifolds are model spaces of particular interest.
Based on [15], a structure theory for Lorentzian homogeneous spaces of finite volume is given by Zeghib [16].
Pseudo-Riemannian homogeneous spaces of arbitrary index were studied by Baues and Globke [2] for solvable Lie groups G. They found that, for solvable G, the finite volume condition implies that the stabilizer H is a lattice in G and that the metric on M is induced by a bi-invariant metric on G. Also, it was observed in [2] that the nil-invariance condition holds for the isometry Lie algebras of finite volume homogeneous spaces, where it appears as a direct consequence of the Borel density theorem. The main result in [2] amounts to showing the surprising fact that any nil-invariant symmetric bilinear form on a solvable Lie algebra g is, in fact, an invariant form (a concise proof is also provided in Appendix B).
By studying metric Lie algebras with nil-invariant symmetric bilinear form, the present work aims to further understand the isometry Lie algebras of pseudo-Riemannian manifolds of finite volume. We will derive a structure theory which allows to completely describe such algebras in index less than 3. In particular, this classification contains all local models for pseudo-Riemannian homogeneous spaces of finite volume of index less than 3.

Main results and structure of the paper
In Section 2, we prove that the orbit maps of isometric actions of Lie groups on pseudo-Riemannian manifolds of finite volume give rise to nil-invariant scalar products on their tangent Lie algebras.
Some basic definitions and properties of metric Lie algebras are reviewed in Section 3.
In favorable cases, nil-invariance of ·, · already implies invariance. For solvable Lie algebras g, this is always the case, as was first shown in [2]. These results are briefly summarized in Section 4. In this section, we will also review the classification of solvable Lie algebras with invariant scalar products of indices 1 and 2. Their properties will be needed further on.

Strong invariance properties
In Section 5, we begin our investigation of nil-invariant symmetric bilinear forms ·, · on arbitrary Lie algebras. For any Lie algebra g, we let g = (k × s) r denote a Levi decomposition of g, where k is semisimple of compact type, s is semisimple of non-compact type and r is the solvable radical of g. For this, recall that k is called of compact type if the Killing form of k is definite and that s is of non-compact type if it has no ideal of compact type. We also write g s = s r.
Our first main result is a strong invariance property for nil-invariant symmetric bilinear forms: Theorem A. Let g be a real finite-dimensional Lie algebra, let ·, · be a nil-invariant symmetric bilinear form on g and ·, · gs the restriction of ·, · to g s . Then: (1) ·, · gs is invariant by the adjoint action of g on g s . (2) ·, · is invariant by g s .
Note that any scalar product on a semisimple Lie algebra k of compact type is already nilinvariant, without any further invariance property required. Therefore, Theorem A is as strong as one can hope for.
Remark. We would like to point out that the proof of Theorem A works for Lie algebras over any field of characteristic zero if the notion of subalgebra of compact type k is replaced by the appropriate notion of maximal anisotropic semisimple subalgebra of g. The latter condition is equivalent to the requirement that the Cartan subalgebras of k do not contain any elements split over the ground field.
We obtain the following striking corollary to Theorem A, or rather to its proof: Corollary B. Let g be a finite-dimensional Lie algebra over the field of complex numbers and ·, · a nil-invariant symmetric bilinear form on g. Then ·, · is invariant.
For any nil-invariant symmetric bilinear form ·, · , it is important to consider its kernel also called the metric radical of g. If ·, · is invariant, then g ⊥ is an ideal of g. If ·, · is nil-invariant, then, in general, g ⊥ is not even a subalgebra of g. Nevertheless, a considerable simplification of the exposition may be obtained by restricting results to metric Lie algebras whose radical g ⊥ does not contain any non-trivial ideals of g. Such metric Lie algebras will be called effective. This condition is, of course, natural from the geometric motivation. Moreover, it is not a genuine restriction, since by dividing out the maximal ideal of g contained in g ⊥ , one may pass from any metric Lie algebra to a quotient metric Lie algebra that is effective.
Theorem A determines the properties of g ⊥ significantly as is shown in the following: Corollary C. Let g be a finite-dimensional real Lie algebra with a nil-invariant symmetric bilinear form ·, · . Assume that the metric radical g ⊥ does not contain any non-trivial ideal of g. Let z(g s ) denote the center of g s . Then The proof of Corollary C can be found in Section 6, which is at the technical heart of our paper. In Subsection 6.1, we start out by studying the totally isotropic ideals in g, and in particular properties of the metric radical g ⊥ . The main part of the proof of Corollary C is given in Subsection 6.2. We then also prove that if in addition ·, · is g ⊥ -invariant, then [g ⊥ , g s ] = 0.
As the form ·, · may be degenerate, it is useful to introduce its relative index. By definition, this is the index of the induced scalar product on the vector space g/g ⊥ . The relative index mostly determines the geometric and algebraic type of the bilinear form ·, · .
For effective metric Lie algebras with relative index 2, we further strengthen Corollary C by showing that, with this additional requirement, g ⊥ does not intersect g s . This is formulated in Corollary 6.21.

Classifications for small index
Section 6 culminates in Subsection 6.5, where we give an analysis of the action of semisimple subalgebras on the solvable radical of g. This imposes strong restrictions on the structure of g for small relative index.
The combined results are summarized in Section 7, leading to the following general structure theorem for the case 2: Theorem D. Let g be a real finite-dimensional Lie algebra with nil-invariant symmetric bilinear form ·, · of relative index 2, and assume that g ⊥ does not contain a non-trivial ideal of g. Then: Examples in Section 8 illustrate that the statements in Theorem D may fail for relative index 3. We specialize Theorem D to obtain classifications of the Lie algebras g in the cases = 1 and = 2. As follows from the discussion at the beginning, these theorems also describe the structure of isometry Lie algebras of pseudo-Riemannian homogeneous spaces of finite volume with index 1 or 2 (real signatures of type (n − 1, 1) or (n − 2, 2), respectively).
Our first result concerns the Lorentzian case: Theorem E. Let g be a Lie algebra with nil-invariant symmetric bilinear form ·, · of relative index = 1, and assume that g ⊥ does not contain a non-trivial ideal of g. Then one of the following cases occurs: where a is abelian and either semidefinite or Lorentzian.
, where a is abelian and definite, and sl 2 (R) is Lorentzian.
This classification of isometry Lie algebras for finite volume homogeneous Lorentzian manifolds is contained in Zeghib's [16, Théorème algébrique 1.11], which uses a somewhat different approach in its proof. Moreover, the list in [16] contains two additional cases of metric Lie algebras (Heisenberg algebra and tangent algebra of the affine group, compare Example 3.3 of the present paper) that cannot appear as Lie algebras of transitive Lorentzian isometry groups, since they do not satisfy the effectivity condition. According to [16], models of all three types (I)-(III) actually occur as isometry Lie algebras of homogeneous spaces G/H, in which case h = g ⊥ is a subalgebra tangent to a closed subgroup H of G.
The algebraic methods developed here also lead to a complete understanding in the case of signature (n − 2, 2): Theorem F. Let g be a Lie algebra with nil-invariant symmetric bilinear form ·, · of relative index = 2, and assume that g ⊥ does not contain a non-trivial ideal of g. Then one of the following cases occurs: is Lorentzian, and r is one of the following: (a) r is abelian and either semidefinite or Lorentzian.
(b) r is Lorentzian of oscillator type.
For the definition of an oscillator algebra, see Example 3.7. The possibilities for r in case (I-c) of Theorem F above are discussed in Section 4.1. Note further that the orthogonality relations of Theorem D part (3) are always satisfied.
Remark. Theorem F contains no information which of the possible algebraic models actually do occur as isometry Lie algebras of homogeneous spaces of index 2. This question needs to be considered on another occasion.
We apply our results to study the isometry groups of simply connected homogeneous pseudo-Riemannian manifolds of finite volume. D'Ambra [5,Theorem 1.1] showed that a simply connected compact analytic Lorentzian manifold (not necessarily homogeneous) has compact isometry group, and she also gave an example of a simply connected compact analytic manifold of metric signature (7,2) that has a non-compact isometry group.
Here, we study homogeneous spaces for arbitrary metric signature. The main result is the following theorem: (1) M is compact.
(2) K is compact and acts transitively on M .
(3) R is abelian. Let A be the maximal compact subgroup of R.
In Section 10, we give examples of isometry groups of compact simply connected homogeneous M with non-compact radical. However, for metric index 1 or 2, the isometry group of a simply connected M is always compact: Theorem H. The isometry group of any simply connected pseudo-Riemannian homogeneous manifold of finite volume with metric index 2 is compact.
As follows from Theorem G, the isometry Lie algebra of a simply connected pseudo-Riemannian homogeneous space of finite volume has abelian radical. This motivates a closer investigation of Lie algebras with abelian radical that admit nil-invariant symmetric bilinear forms in Section 9. In this direction, we prove: Theorem I. Let g be a Lie algebra whose solvable radical r is abelian. Suppose that g is equipped with a nil-invariant symmetric bilinear form ·, · such that the metric radical g ⊥ of ·, · does not contain a non-trivial ideal of g. Let k × s be a Levi subalgebra of g, where k is of compact type and s has no simple factors of compact type. Then g is an orthogonal direct product of ideals with g 1 = k a, g 2 = s 0 , g 3 = s 1 s * 1 , where r = a × s * 1 and s = s 0 × s 1 are orthogonal direct products, and g 3 is a metric cotangent algebra. The restrictions of ·, · to g 2 and g 3 are invariant and non-degenerate. In particular, For the definition of metric cotangent algebra, see Example 8.1. We call an algebra g 1 = k a with k semisimple of compact type and a abelian a Lie algebra of Euclidean type. By Theorem G, isometry Lie algebras of compact simply connected pseudo-Riemannian homogeneous spaces are of Euclidean type. However, not every Lie algebra of Euclidean type appears as the isometry Lie algebra of a compact pseudo-Riemannian homogeneous space. In fact, this is the case for the Euclidean Lie algebras e n = so n R n with n = 3.
Theorem J. The Euclidean group E n = O n R n , n = 1, 3, does not have compact quotients with a pseudo-Riemannian metric such that E n acts isometrically and almost effectively.
Note that E n acts transitively and effectively on compact manifolds with finite fundamental group, as we remark at the end of Section 9.

Notations and conventions
The identity element of a group G is denoted by e. We let G • denote the connected component of the identity of G.
Let H be a subgroup of a Lie group G. We write Ad g (H) for the adjoint representation of H on the Lie algebra g of G, to distinguish it from the adjoint representation Ad(H) on its own Lie algebra h.
If V is a G-module, then we write V G = {v ∈ V | gv = v for all g ∈ G} for the module of G-invariants. Similarly, V g = {v ∈ V | Xv = 0 for all X ∈ g} for a g-module.
The centralizer and the normalizer of h in g are denoted by z g (h) and n g (h), respectively. The center of g is denoted by z(g). We use similar notation for Lie groups.
If g 1 and g 2 are two Lie algebras, the notation g 1 × g 2 denotes the direct product of Lie algebras. The notations g 1 + g 2 and g 1 ⊕ g 2 are used to indicate sums and direct sums of vector spaces.
The solvable radical r of g is the maximal solvable ideal of g. The semisimple Lie algebra f = g/r is a direct product f = k × s of Lie algebras, where k is a semisimple Lie algebra of compact type, meaning that the Killing form of k is definite, and s is semisimple without factors of compact type.
For any linear operator ϕ, ϕ = ϕ ss + ϕ n denotes its Jordan decomposition, where ϕ ss is semisimple, and ϕ n is nilpotent. Further notation will be introduced in Section 3.

Isometry Lie algebras
Let (M, g) be a pseudo-Riemannian manifold of finite volume, and let G ⊆ Iso(M, g) be a Lie group of isometries of M . Identify the Lie algebra g of G with a subalgebra of Killing vector fields on (M, g). Let S 2 g * denote the space of symmetric bilinear forms on g, and let The adjoint representation of G on g induces a representation : G → GL(S 2 g * ).
Theorem 2.1. Let A be the real Zariski closure of (G) in the group GL(S 2 g * ). Let p ∈ M . Then the bilinear form Φ p is invariant by all unipotent elements in A.
Proof. Note that the above Gauß map Φ is equivariant with respect to , since G acts by isometries on M . The pseudo-Riemannian metric on M defines a finite G-invariant measure on M .
Since the claim clearly holds on totally isotropic G-orbits, we may in the following assume that all orbits of G are non-isotropic, that is, Φ p = 0 for all p ∈ M .
Put V = S 2 g * . For a subset W ⊆ V \0, let W denote its image in the projective space P(V ). Similarly, for subsets in GL(V ) and their image in the projective linear group PGL(V ).
The finite G-invariant measure on M induces a finite G-invariant measure ν on the projective space P(V ) with support supp ν = Φ(M ) ⊂ P(V ). Let PGL(V ) ν denote the stabilizer of ν in the projective linear group. This is a real algebraic subgroup of PGL(V ), cf. [ Since PGL(V ) ν is real algebraic, the image of A in PGL(V ) is contained in PGL(V ) ν . Choose W i such that Φ p ∈ W i . Let u ∈ A be a unipotent element. Since the restriction of u to PGL(W i ) is unipotent and it is contained in a compact subset of PGL(W i ), it must be the identity of W i . This implies u · Φ p = Φ p .
In terms of Definition 3.1 below, this implies the following: Corollary 2.2. For p ∈ M , let ·, · p denote the symmetric bilinear form induced on the Lie algebra g of G by pulling back g p along the orbit map g → g · p. Then ·, · p is nil-invariant and its kernel contains the Lie algebra g p of the stabilizer G p of p in G. If G acts transitively on M , then the kernel of ·, · p equals g p .

Metric Lie algebras
Let g be a finite-dimensional real Lie algebra with a symmetric bilinear form ·, · . The pair (g, ·, · ) is called a metric Lie algebra † .
Let h be a subalgebra of g. The restriction of ·, · to h will be denoted by ·, · h . The form This is the maximal subalgebra of g under which ·, · is invariant. If ·, · is g-invariant, we simply say ·, · is invariant. The kernel of ·, · is the subspace It is also called the metric radical for (g, ·, · ). It is an invariant subspace for the Lie brackets with elements of inv(g, ·, · ), and, if ·, · is invariant, then g ⊥ is an ideal in g.

Nil-invariant bilinear forms
Let Inn(g) z denote the Zariski closure of the adjoint group Inn(g) in Aut(g).
for all nilpotent elements ϕ of the Lie algebra of Inn(g) z .
In particular, (3.2) holds for the nilpotent parts ϕ = ad(Y ) n of the Jordan decomposition of the adjoint representation of any Y ∈ g.

Index of symmetric bilinear forms
Let ·, · be a symmetric bilinear form on a finite-dimensional vector space so that (V ) is the index of the non-degenerate bilinear form induced by ·, · on V /V ⊥ . We call the relative index of V (or ·, · ).
When there is no ambiguity about the space V , we simply write μ = μ(V ) and = (V ). We then say that V is of index type. In particular, for = 1, we say V is of Lorentzian type. We call V Lorentzian if μ = = 1.
If ·, · is non-degenerate, that is, if V ⊥ = 0, then we call ·, · a scalar product on V . We say that the scalar product ·, · is definite if μ = 0.
Let W ⊆ V be a vector subspace. We say W is definite, Lorentzian, of relative index (W ) or of index μ(W ), respectively, if the restriction ·, · W is. Observe further that μ(W ) μ(V ) and (W ) (V ).

Examples of metric Lie algebras
Example 3.2. Consider R n with a scalar product ·, · represented by the matrix ( I n−s 0 0 −Is ), where s n − s. Then ·, · has index s, and we write R n s for (R n , ·, · ). If we take R n to be an abelian Lie algebra, together with ·, · , it becomes a metric Lie algebra denoted by ab n s .
The Heisenberg algebra occurs naturally in the construction of Lie algebras with invariant scalar products.
Example 3.3. Let (·, ·) be a Hermitian form on C n . Define the Heisenberg algebra h 2n+1 as the vector space C n ⊕ z, where z = span{Z}, with Lie brackets defined by for any X, Y ∈ C n . Thus, h 2n+1 is a real 2n + 1-dimensional two-step nilpotent Lie algebra with one-dimensional center (as such it is unique up to isomorphism of Lie algebras). Equip C n with the bilinear product ·, · = Re(·, ·). Declaring z to be perpendicular to h 2n+1 turns h 2n+1 into a metric Lie algebra, whose relative index (h 2n+1 ) is determined by the index of the Hermitian form.
where J acts by multiplication with the imaginary unit on C n . Given any metric on h 2n+1 as in Example 3.3, an invariant scalar product ·, · of index (h 2n+1 ) + 1 on osc is obtained by requiring J, Z = 1 and C n ⊥ J. Example 3.4 is an important special case of the following construction: Example 3.5. Given ψ ∈ so n−s,s define the oscillator algebra as follows. On the vector space g = d ⊕ ab n s ⊕ j with d = span{D}, j = span{Z}, define a Lie product by declaring: where X, Y ∈ ab n s . Next extend the indefinite scalar product on ab n s to g by D, D = Z, Z = 0, D, Z = 1, (a ⊕ j) ⊥ ab n s . Then ·, · is an invariant scalar product of index s + 1 on g. The Lie algebra osc(ψ) is solvable. It is nilpotent if and only if ψ is nilpotent. If ψ is a k-step nilpotent operator, then g is a k-step nilpotent algebra. If ψ is not zero, then the ideal h = ab n s ⊕ j is of Heisenberg type (that is, nilpotent with one-dimensional commutator [h, h] = j).

Invariant Lorentzian scalar products. The main building blocks for metric Lie algebras with invariant Lorentzian scalar products are obtained by:
Example 3.6. The Killing form on sl 2 (R) is an invariant Lorentzian scalar product. In fact, all semisimple Lie algebras with an invariant Lorentzian scalar product are products of sl 2 (R) by simple factors of compact type.
Example 3.7. For ψ ∈ so n , the oscillator algebra osc(ψ) is Lorentzian. We say that such a metric Lie algebra is Lorentzian of oscillator type.
Remark. Classification of Lie algebras with invariant Lorentzian scalar products was derived by Medina [10] and by Hilgert and Hofmann [7]. It can be deduced that algebras of oscillator type are the only non-abelian solvable Lie algebras which admit an invariant Lorentzian scalar product. This is also a direct consequence of the reduction theory of solvable metric Lie algebras, see Section 4.

Review of the solvable case
The first two authors studied nil-invariant symmetric bilinear forms on solvable Lie algebras in [2]. The main result [2, Theorem 1.2] is: Theorem 4.1. Let g be a solvable Lie algebra and ·, · a nil-invariant symmetric bilinear form on g. Then ·, · is invariant. In particular, g ⊥ is an ideal in g.
An important tool in the study of (nil-)invariant products ·, · on solvable g is the reduction by a totally isotropic ideal j in g. Since ·, · is invariant, j ⊥ is a subalgebra. Therefore, we can consider the quotient Lie algebra Since j is totally isotropic, g inherits a non-degenerate symmetric bilinear form from j ⊥ that is (nil-)invariant as well. The metric Lie algebra (g, ·, · ) is called the reduction of (g, ·, · ) by j. Reduction by j decreases the index of ·, · . Let n be the nilradical of g. The ideal is a characteristic totally isotropic ideal in g, whose orthogonal space j ⊥ 0 is also an ideal in g and contains j 0 in its center. Then j 0 = 0 if and only if g is abelian. In particular, g is abelian if ·, · is definite. This implies [2, Proposition 5.4]: Proposition 4.2. Let (g, ·, · ) be a solvable metric Lie algebra with nil-invariant symmetric bilinear form ·, · . After a finite sequence of reductions with respect to totally isotropic and central ideals, (g, ·, · ) reduces to an abelian metric Lie algebra.
The proposition is useful in particular to derive properties of solvable metric Lie algebras of low index.

Invariant scalar products of index 2
Example 4.3. Let ψ ∈ so n,1 . Then the oscillator algebra osc(ψ) as defined in Example 3.5 is of index = 2.
The three families of Lie algebras in Examples 4.3, 4.4 and 4.5 were found by Kath and Olbrich [9] to contain all indecomposable non-simple metric Lie algebras with invariant scalar product of index 2. Thus, we note: Proposition 4.6. Any solvable metric Lie algebra with invariant scalar product of index 2 is obtained by taking direct products of metric Lie algebras in Examples 3.7 and 4.3-4.5 or abelian metric Lie algebras.
We use this to derive the following particular observation, which will play an important role in Section 6.5. An ideal in a metric Lie algebra (g, ·, · ) is called characteristic if it is preserved by every skew derivation of g.
Proposition 4.7. Let g be a solvable Lie algebra with invariant bilinear form ·, · of index μ 2. Then (g, ·, · ) has a characteristic ideal q that satisfies: Proof. It is easily checked that the characteristic ideal q = [g, g] + z(g) of g satisfies (1) and (2) for the Examples 3.7 and 4.3-4.5, and for products of oscillators as in Example 3.7. Hence, the proposition is satisfied for all invariant scalar products of index = μ 2.
Suppose now that ·, · is degenerate and = 0. Then g/g ⊥ inherits a definite invariant scalar product. Hence, g/g ⊥ is abelian, and [g, g] ⊆ g ⊥ . Since dim g ⊥ μ 2, it follows that q = g has the required properties.
In the following, n denotes the nilradical of the Lie algebra g.
Corollary 4.8. Let g be a solvable metric Lie algebra which admits an invariant scalar product of index 2. If g is not nilpotent, then

Nil-invariant symmetric bilinear forms
Let g be a finite-dimensional real Lie algebra with solvable radical r. Let be a Levi decomposition, where k is semisimple of compact type and s is semisimple without factors of compact type. Furthermore, we put g s = s r.
Note that g s is a characteristic ideal of g.
The purpose of this section is to show: Theorem A. Let ·, · be a nil-invariant symmetric bilinear form on g, and let ·, · gs denote the restriction of ·, · to g s . Then: (1) ·, · gs is invariant by the adjoint action of g on g s .
The proof of Theorem A begins with a few auxiliary results.
Lemma 5.1. Let s ⊆ g be a semisimple subalgebra of non-compact type. Then the subalgebra generated by all X ∈ s, such that ad(X) : g → g is nilpotent, is s.
Proof. Call X ∈ s nilpotent if ad(X) : s → s is nilpotent. Since, for every representation of s, nilpotent elements are mapped to nilpotent operators, it is sufficient to prove the statement for s = g. So, let s 0 be the subalgebra of s generated by all nilpotent elements. Since the set of all nilpotent elements is preserved by every automorphism of s, it follows that s 0 is an ideal. Therefore, the semisimple Lie algebra s 1 = s/s 0 does not contain any nilpotent elements. Let a be a Cartan subalgebra of s 1 , and a s the subspace consisting of elements X ∈ a, where ad(X) is split semisimple (that is, diagonalizable over R). The weight spaces for the non-trivial roots of a s consist of nilpotent elements of s 1 . Since, by construction, s 1 has no nilpotent elements, this implies that a has no elements split over R. This, in turn, implies that s 1 is of compact type (cf. Borel [3, § 24.6(c)]). By assumption, s is of non-compact type, so s 1 must be trivial.
Lemma 5.2. Let n be the nilradical of g. Then ·, · is invariant by s n.
Proof. Since ·, · is nil-invariant, inv(g, ·, · ) contains all X such that the operator ad(X) : g → g is nilpotent. In particular, n is contained in inv(g, ·, · ). Since s is of non-compact type, the subalgebra generated by all X ∈ s with ad(X) nilpotent is s, see Lemma 5.1. Therefore, also s ⊆ inv(g, ·, · ).
Recall that any derivation ϕ of the solvable Lie algebra r satisfies ϕ(r) ⊆ n (Jacobson [8,Theorem III.7]). In particular, if ϕ is semisimple, there exists a decomposition r = a + n into vector subspaces, where ϕ(a) = 0. Similarly, for any subalgebra h of g acting reductively on r, we have r = r h + n, where [h, r h ] = 0. Lemma 5.3. Let h be a subalgebra of inv(g, ·, · ), and let g h be the maximal trivial submodule for the adjoint action of h on g.
Proof of Theorem A, part (1). Since s acts reductively on g, we have g = g s + g s . Therefore, by Lemma 5 By Theorem 4.1, the restriction of ·, · to the solvable Lie algebra generated by r and X is invariant on that subalgebra. Hence, Proof. Let X ∈ g f , Y ∈ g s and K ∈ f. Since g s is an ideal in g, we may write g s = (g s ∩ ker ad(X) ss ) + ad(X)g s .
Suppose first that Y ∈ ker ad(X) ss . Then we get The latter term vanishes, since K ∈ ker ad(X) ⊆ ker ad(X) n .
Proof of Theorem A, part (2). By Lemma 5.2, ·, · is invariant by s + n. Since g s = s + r k + n, to prove that ·, · is g s -invariant, it suffices to show that ad(X) is skew for all X ∈ r k . By part (1), the restriction of ad(X) to the ideal g s is skew. Hence, it remains to show that Note that the right term is zero because of Lemma 5.4. Example 5.6 (Nil-invariant products on semisimple Lie algebras). Let g = k × s be semisimple, where k is an ideal of compact type and s is of non-compact type. For any nil-invariant bilinear form ·, · , (g, ·, · ) = (k, ·, · k ) × (s, ·, · s ) decomposes as a direct product of metric Lie algebras, where ·, · s is invariant.

Totally isotropic ideals and metric radicals
Let g be a finite-dimensional real Lie algebra with nil-invariant symmetric bilinear form ·, · and subalgebras k, s, r, g s as in Section 5. We let denote the relative index of ·, · (which is the index of the non-degenerate bilinear form induced by ·, · on g/g ⊥ ).

Transporter algebras
For any subspaces U ⊆ V of g and any subalgebra q of g, define Clearly, n q (V, U ) is a subalgebra of q. Also, [q, V ] ⊆ U if and only if n q (V, U ) = q.
Suppose that b ⊆ g s is a totally isotropic ideal of g contained in g s . Then consider By Theorem A part (2), ·, · is invariant by g s . Therefore, b 0 is an ideal of g s . For any subalgebra q of g, define the transporter subalgebra for b in q as Lemma 6.1 (Transporter lemma). For q, b, b 0 as above, we have . This shows the equivalence of X ⊥ [g, b] and X ∈ n q (b, b 0 ). Hence, equation (6.2) holds. As Remark. Equality holds in (6.3) if and only if dim q + b ⊥ = dim g.
The following relations between transporters are satisfied: gs (b, b 0 ). This shows (2).
Corollary 6.4. Assume that g ⊥ does not contain any non-trivial ideal of g. Then every totally isotropic ideal b of g contained in g s is abelian.
The case of a large transporter in k has particularly strong consequences: Proposition 6.5. Assume that n k (b, b 0 ) = k. Then: If furthermore g ⊥ does not contain any non-trivial ideal of g, then: Proof. Since, by Theorem A, ·, · is invariant by g s , [g s , 6.1.2. Metric radical of g s . In the following, consider the special case: b = g ⊥ s ∩ g s . Thus, b is totally isotropic and it is the metric radical of g s (with respect to the induced metric ·, · gs ). By Theorem A, ·, · gs is invariant by g. Therefore, b is an ideal in g. Moreover, is an ideal in g s .
Since [g s , b] is an ideal in g, we deduce: Corollary 6.7. If g ⊥ does not contain a non-trivial ideal of g, then g ⊥ s ∩ g s ⊆ z(g s ). In particular, g ⊥ ∩ g s ⊆ z(g s ).
The following strengthens Proposition 6.5 for b = g ⊥ s ∩ g s and b 0 = g ⊥ ∩ g s : Proposition 6.8. Assume that n k (b, b 0 ) = k. Then: In particular, g ⊥ ∩ g s is an ideal in g. If furthermore g ⊥ contains no non-trivial ideal of g, then: (1) and (2) follow, and also (3), since [g ⊥ s , g s ] ⊆ b 0 , by Lemma 6.6.
Remark. It is not difficult to see (compare Lemma 6.9 below) that the centralizer of g s in g is z g (g s ) = z k (g s ) × z(g s ).
6.2. Metric radical of g Proof. Let g = f r (where f ⊇ k is a semisimple subalgebra, and r the maximal solvable ideal of g) be a Levi decomposition of g. Assume first that W is an irreducible g-module. Then the action of r on W is reductive and commutes with f. Since the image of f in gl(W ) has trivial center, the claim of the lemma follows in this case. For the general case, consider a Hölder sequence of submodules W ⊇ W 1 ⊇ . . . ⊇ W k = 0 such that the g-module W i /W i+1 is irreducible. The above implies that, for any Z = K + X ∈ c, where K ∈ f and X ∈ r, K (and X) act trivially on W i /W i+1 . Since K ∈ k is semisimple on W , this implies that K acts trivially on W . That is, K ∈ c ∩ k and therefore also X ∈ c ∩ r. Proposition 6.10. If g ⊥ does not contain a non-trivial ideal of g, then Proof. By Lemma 6.6 , which is the centralizer of W . In view of our assumption on ideals in g ⊥ , observe that [c, g s ] ⊆ b ⊆ z(g s ) ⊆ r by Corollary 6.7. Now [c, g s ] ⊆ r implies that c is contained in k + r. Therefore, Lemma 6.9 applies, showing g ⊥ s ⊆ n k (g s , g s ∩ g ⊥ s ) + n r (g s , g s ∩ g ⊥ s ). Since [r, z(g s )] = 0, it follows that n r (g s , z(g s )) ⊆ n n (g s , z(g s )). Hence, (1) holds.
To prove (2), suppose Z = K + X ∈ n(g s , g s ∩ g ⊥ ), where K ∈ k, X ∈ r. By (1), K ∈ c = n(g s , g s ∩ g ⊥ s ). Since K acts as a semisimple derivation on g s , we can decompose , Y for all Y ∈ g. By Lemma 6.6, [X, v] ∈ g ⊥ . This implies [K, w] = [K, v] ∈ g ⊥ . It also follows that [X, w] ∈ g ⊥ . Lemma 6.11. Let j = n n (g s , z(g s ) ∩ g ⊥ ). Then j is an ideal in g.
These considerations yield the following important property of g ⊥ : Theorem 6.12. Suppose that g ⊥ does not contain a non-trivial ideal of g. Then g ⊥ ⊆ n k g s , z(g s ) ∩ g ⊥ + z(g s ).
Proof. Consider the ideal j, as defined in Lemma 6.11. By (2) of Proposition 6.10, we have g ⊥ ⊆ n k (g s , z(g s ) ∩ g ⊥ ) + j. Since j is an ideal in g, so is [j, g s ]. Since [j, g s ] ⊆ g ⊥ , the assumption on ideals in g ⊥ implies that [j, g s ] = 0. It follows that j is contained in z(g s ).
6.2.1. Invariance by g ⊥ . We shall be interested in nil-invariant bilinear forms ·, · on g induced by pseudo-Riemannian metrics on homogeneous spaces. In this case, ·, · is invariant by the stabilizer subalgebra g ⊥ . We can then further sharpen the statement of Corollary C. Proposition 6.13. Let g and ·, · be as in Corollary C. If in addition ·, · is g ⊥ -invariant, then The proof is based on the following immediate observations: Lemma 6.14. Suppose that ·, · is g ⊥ -invariant. Then [[k, g ⊥ ], g s ] ⊆ g ⊥ ∩ g s . Proof of Proposition 6.13. Let k 0 be the image of g ⊥ under the projection homomorphism g → k. Note that by Corollary C, [g ⊥ , g s ] = [k 0 , g s ]. Let q ⊆ k be the subalgebra generated by m = k 0 + [k, k 0 ] and consider V = g s as a module for q. Since q is an ideal of k, [q, V ] is a submodule for k, that is, [k, [q, V ]] ⊆ [q, V ]. By Lemmas 6.14, 6.15 and Corollary C, we have Since g ⊥ contains no non-trivial ideals of g by assumption, we conclude that j = 0. Lemma 6.17. Let k be a semisimple Lie algebra of compact type and l a subalgebra of k. Then either l = k or m = codim k l > 1. Assume further that l does not contain any non-trivial ideal of k. Then, up to conjugation by an automorphism of k:

Transporter in k and low relative index
(1) if m = 2, then k = so 3 and l = so 2 , (2) if m = 3, one of the following holds: (a) k = so 3 and l = 0, (b) k = so 3 × so 3 , l is the image of a diagonal embedding so 3 → so 3 × so 3 .
Proof. As an ad k (l)-module, k = l ⊕ w for a submodule w. For this, note that any Lie subalgebra of k acts reductively, since k is of compact type.
Suppose codim k l = 1, that is, w is one-dimensional. Then [w, w] = 0 and it follows that w is also an ideal of k. A one-dimensional ideal cannot exist, since k is semisimple. It follows that codim k l > 1.
Since k = l ⊕ w, the kernel of the adjoint action of l on w is an ideal in k. Assume further that l contains no non-trivial ideals of k. Then l acts faithfully on w.
For m = 2, this means l = so 2 and dim k = m + dim l = 3. Hence, k = so 3 . For m = 3, l embeds into so 3 . If l = 0, we have dim k = 3 and thus k = so 3 . Otherwise, either l = so 2 or l = so 3 . In the first case, dim k = 4. Since there is no four-dimensional simple Lie algebra, this is not possible. In the latter case, dim k = 6. This leaves k = so 3 × so 3 (being isomorphic to so 4 ) as the only possibility. Since l is not an ideal of k, l projects injectively onto both factors of k. It follows that, up to automorphism of k, l is the image of an embedding so 3 → so 3 × so 3 , X → (X, X).

Totally isotropic ideals and low relative index.
Let b be any totally isotropic ideal of g contained in g s and put b 0 = b ∩ g ⊥ . Proof. Put l = n k (b, b 0 ) and m = codim k l. By Lemma 6.1, l = k ∩ [g, b] ⊥ and m . Assume now that m 1. According to Lemma 6.17, the case m = 1 never occurs. Hence, in this case, we have m = 2.
Let i ⊆ l be the maximal ideal of k contained in l. Using Proposition 6.3, we see that there exists an ideal b 1 of g, such that induces a skew pairing on k × U , such that U ⊥ = l. Since we have i ⊥ [g, b], this shows that ·, · restricted to k × [g, b] descends to a skew pairing If m = 2, then by Lemma 6.17, k/i = so 3 and l/i = so 2 . By Corollary A.6, either the skew pairing ·, · in (6.4) is zero (that is, In the second case, l = i, a contradiction to l/i = so 2 . Therefore, m = 0. Combining with Proposition 6.5(1), we arrive at: Corollary 6.19. If 2 then, for any totally isotropic ideal b of g contained in g s , g ⊥ ∩ b is an ideal in g. In particular, g ⊥ ∩ g s is an ideal in g.
The following now summarizes our results on totally isotropic ideals in case 2: Corollary 6.20. Assume that g ⊥ does not contain any non-trivial ideal of g and that 2. Then, for any totally isotropic ideal b of g contained in g s , Furthermore, the following hold: Proof. Since 2, according to Proposition 6.18 n k (b, b 0 ) = k. Thus, (1) holds due to part (2) of Proposition 6.5. Now (2)-(4) are consequences of Proposition 6.8.
Combining with Theorem 6.12, we also obtain: Corollary 6.21. Assume that g ⊥ does not contain any non-trivial ideal of g and that 2. Then g ⊥ is contained in z(g s ) × k and g ⊥ ∩ g s = 0.
Remark. As Example 8.2 shows, these conclusions do not necessarily hold if 3.

Metric radicals of the characteristic ideals
This section serves to clarify the relations between the metric radicals of g s , r and n, where n denotes the nilradical of r. The lemma is clearly implied by: Lemma 6.23. Let j ⊆ g s be an ideal in g. Then the following hold: Proof. Since ·, · restricted to g s is g-invariant by Theorem A, j ⊥ ∩ g s is an ideal in g. It follows that j ⊥ ∩ j is an ideal. Hence, (1) holds. Now (2) follows using the above remark with a = j, c = g and b = j.
6.4.1. Radicals in effective metric Lie algebras. For all following results, we shall also require that the metric Lie algebra (g, ·, · ) is effective. That is, we assume for now that g ⊥ does not contain any non-trivial ideal of g. Lemma 6.24. Let i, j ⊆ g s be ideals in g. Then: Proof. By Lemma 6.23(1), [j, j ⊥ ∩ g s ] is an ideal of g and contained in g ⊥ . Since g ⊥ does not contain any non-trivial ideal of g, [j, j ⊥ ∩ g s ] = 0. Hence, (1) holds. Under the assumption of (2), this means i ⊥ ∩ i ⊆ z(i) ⊆ j. Since also i ⊥ ⊆ j ⊥ , (2) follows.
The next result somewhat strengthens Corollary 6.7.
Proposition 6.25. The following hold: Proof. By Lemma 6.23(1), j ⊥ ∩ g s is an ideal of g for any ideal j of g contained in g s . Then for any ideal i of g, In the view of Lemma 6.22, (1)-(3) follow.
We can deduce from (2) of Proposition 6.25 the equalities Also (3) of Proposition 6.25 shows that Moreover, using nil-invariance of ·, · and Corollary 5.5(1), the above yield Thus there is a tower of totally isotropic ideals of g contained in z(r): (6.9)

Actions of semisimple subalgebras on the solvable radical
Let q be a subalgebra of g. We call the subspace W ⊆ g a submodule for q if [q, W ] ⊆ W . In the following, we let f ⊆ g denote a semisimple subalgebra of g. As usual, we decompose f = k × s, where k is an ideal of compact type and s has no factor of compact type.  Lemma 6.27. Let q be a subalgebra of g, and let W ⊆ r be a submodule for q. Then the following hold for l = q ∩ [W, W ] ⊥ : (1) l is a subalgebra, and [l, W ] ⊆ W ⊥ .
Assume further that q acts reductively on W . Then: If q = f is semisimple and dim[W 1 , W 1 ] 2, then: To finish the proof of (1), assume that . This shows that l is a subalgebra.
Next we show (2). Since q acts reductively on W , Finally, if q acts reductively, there is a decomposition into submodules W = (W ∩ W ⊥ ) ⊕ W . Correspondingly, K ∈ l if and only if [K, W ] = 0. This shows that l is an ideal in q. Hence, (3) holds.
If f is semisimple, then f acts reductively on W . By part (3), l = f ∩ [W, W ] ⊥ is an ideal of f. Since dim[W, W ] 2, it is an ideal of codimension at most 2. Since f is semisimple, this implies l = f. Hence, (4) holds. Now (4) together with (1) implies that [f, W ] ⊆ W ⊥ is totally isotropic.
For any subspace W of g, recall that μ(W ) denotes the index of W . We are ready to give the main result of this subsection. Proof. We have μ(r) μ(g s ) 2. Thus, Proposition 4.7 implies that there exists an ideal q of g with dim[q, q] 2, and the codimension of q in r is at most 2. Since μ(g s ) 2, [k × s, r] = 0, by Lemma 6.28. This also implies s ⊥ r (compare Lemma 6.26).
As a consequence, we further get: Lemma 6.30. Suppose μ(g s ) 2. Then the following hold: (1) s is non-degenerate.
Proof. Note that dim s ∩ s ⊥ μ(g s ) 2. Since ·, · s is invariant, s ∩ s ⊥ is an ideal in s. We conclude that s ∩ s ⊥ = 0. This shows (1).

Lie algebras with nil-invariant scalar products of small index
Partially summarizing the results from Proposition 6.29 and Corollary 6.21, we obtain a first structure theorem for metric Lie algebras of relative index 2.
Theorem D. Let g be a real finite-dimensional Lie algebra with nil-invariant symmetric bilinear form ·, · of relative index 2, and assume that g ⊥ does not contain a non-trivial ideal of g. Then: (1) The Levi decomposition (5.1) of g is a direct sum of ideals: g = k × s × r.
We will now study the cases = 0, = 1 and = 2 individually.

Semidefinite nil-invariant products
Let ·, · be a nil-invariant symmetric bilinear form on g.
Proposition 7.1. If ·, · is semidefinite (the case = 0), then Moreover, if g ⊥ does not contain any non-trivial ideal of g, then: (2) g = k × r and r is abelian.
(3) The ideal r is definite.
Proof. According to Theorem A, nil-invariance implies that g s acts by skew derivations on g and on g/g ⊥ . By assumption, ·, · induces a definite scalar product on the vector space g/g ⊥ . Recall that a definite scalar product does not allow nilpotent skew maps. Therefore, [s + n, g] ⊆ g ⊥ . Similarly, for X ∈ r, ad(X) n (g) ⊆ g ⊥ and thus also [r, r] ⊆ [r, n] + g ⊥ ⊆ g ⊥ . Moreover, [r, k × s] = [n, k × s] ⊆ g ⊥ . This shows (1), while (2) and (3) follow immediately, taking into account Theorem D.

Classification for relative index 2
Now we specialize Theorem D to the two cases = 1 and = 2 to obtain classifications of the Lie algebras with nil-invariant symmetric bilinear forms in each case.
Theorem E. Let g be a Lie algebra with nil-invariant symmetric bilinear form ·, · of relative index = 1, and assume that g ⊥ does not contain a non-trivial ideal of g. Then one of the following cases occurs: where a is abelian and either semidefinite or Lorentzian. (II) g = k × r, where r is Lorentzian of oscillator type. (III) g = k × sl 2 (R) × a, where a is abelian and definite, sl 2 (R) is Lorentzian and (a × k) ⊥ sl 2 (R).
Proof. By Theorem D, g ⊥ ∩ r = 0. Hence, r is a subspace of index μ(r) 1. If s = 0, then by Lemma 6.30, s is non-degenerate, so that (s) = 1, as s is of non-compact type. Hence, s = sl 2 (R). Moreover, (r) = 0, that is, r is definite and therefore abelian. The orthogonality is given by Lemma 6.30. This is case (III).
Otherwise, s = 0. If r is semidefinite, then [r, r] ⊆ r ⊥ ∩ r by Proposition 7.1. By Lemma 6.30(2), this implies [r, r] ⊆ g ⊥ ∩ r = 0. Hence, r is abelian. This is the first part of case (I). Assume that r is of Lorentzian type. Then r is non-degenerate since μ(r) 1. By the classification of invariant Lorentzian scalar products (see remark following Example 3.7), r is either abelian or contains a metric oscillator algebra. These are the second part of case (I) or case (II), respectively.
Theorem F. Let g be a Lie algebra with nil-invariant symmetric bilinear form ·, · of relative index = 2, and assume that g ⊥ does not contain a non-trivial ideal of g. Then one of the following cases occurs: (a) g = r × k, where r is one of the following: (a) r is abelian.
(b) r is Lorentzian of oscillator type.
(c) r is solvable but non-abelian with invariant scalar product of index 2. Proof. Write s = μ(s) and r = μ(r). By Theorem D, g ⊥ ∩ g s = 0. By Lemma 6.30, this implies s + r 2. Moreover, s is non-degenerate and thus has index s 2. First, assume s = 0, and therefore s = 0, and r 2. For r = 2, the following possibilities arise: r is non-degenerate with relative index (r) = 2. This case falls into (I-a) or (I-c). Next, r can be degenerate with (r) = 1, in which case it is either abelian or of oscillator type, the latter yielding part (I-b). In the remaining case (r) = 0, r is semidefinite. As in the proof of Theorem E, this implies that r = a is abelian. This completes case (I).
Assume s = 2 and r = 0. Then s = sl 2 (R) × sl 2 (R) and r is definite and abelian. The orthogonality is Lemma 6.30(3). This is case (II) Now assume s = 1 and s = sl 2 (R), r 1. This yields the two possibilities for r in case (III).
Note that the possible Lie algebras r for case (I-c) of Theorem F above are discussed in Section 4.1.

Further examples
The examples in this section show that the properties of nil-invariant symmetric bilinear forms with relative index 2 given in Theorem D do not hold for higher relative indices. Let k, s, r and g s be as in the previous sections.
The following standard construction for a Lie algebra with an invariant scalar product (cf. Medina [10]) shows that in general g does not have to be a direct product of Lie algebras k, s and r, and that s does not have to be orthogonal to r. Example 8.1 (Metric cotangent algebras). Let g be a Lie algebra of dimension n, and let ad * denote the coadjoint representation of g on its dual vector space g * , and consider the Lie algebra g = g ad * g * . The dual pairing defines an invariant scalar product on g, where X i ∈ g and ξ i ∈ g * . The index of ·, · is n. We call such a g a metric cotangent algebra. For example, if we choose g = sl 2 (R), then the index is 3, and k = 0, s = sl 2 (R) and g has abelian radical r = sl 2 (R) * ∼ = R 3 . In particular, s is not orthogonal to r and [s, r] = 0.
The next example shows that for relative index = 3, the transporter algebra l of b = r ⊥ ∩ r in q = k (see Section 6.3.1) can be trivial, and as a consequence, g ⊥ ∩ r is not an ideal in g. This contrasts the situation for 2, compare Corollary 6.19.
Example 8.2. Let k = so 3 , and let r = so 3 ⊕ so 3 , considered as a vector space. We write so l 3 and so r 3 to distinguish the two summands of r, and for an element X ∈ so 3 , we write X l = (X, 0) ∈ so l 3 and X r = (0, X) ∈ so r 3 . Let so 3 be the diagonal embedding of so 3 in r.
This makes r into a Lie algebra module for k, and we can thus define a Lie algebra g = k r for this action, taking r as an abelian subalgebra. Observe also that so r 3 is the center of g. Let κ denote the Killing form on so 3 . We define a symmetric bilinear form ·, · on g by requiring T, X l 1 + X r 2 = κ(T, X 1 ) − κ(T, X 2 ), k ⊥ k, r ⊥ r for all T ∈ k, X l 1 + X r 2 ∈ so 3 . The adjoint operators of elements of r are skew-symmetric for ·, · . In fact, we have, for all X, Y ∈ r, Z ∈ g, and for T, T ∈ k, by ( * ), So ·, · is indeed a nil-invariant form on g, and, since r ⊥ = r, g ⊥ = k ⊥ ∩ r = so 3 and g ⊥ s ∩ g s = r ⊥ ∩ r = r = so l 3 ⊕ so r 3 . In particular, the index of ·, · is μ = 6 and the relative index is = 3. Note that ·, · is not invariant, as g ⊥ is not an ideal in g.
Remark. The construction in Example 8.2 works if we replace so 3 by any other semisimple Lie algebra f = k of compact type. However, if f is not of compact type, then the resulting bilinear form ·, · will not be nil-invariant. Geometrically, this means that ·, · cannot come from a pseudo-Riemannian metric on a homogeneous space G/H of finite volume, where G is a Lie group with Lie algebra g.

Metric Lie algebras with abelian radical
In this section, we study finite-dimensional real Lie algebras g whose solvable radical r is abelian and which are equipped with a nil-invariant symmetric bilinear form ·, · .

Abelian radical
The Lie algebras with abelian radical and a nil-invariant symmetric bilinear form decompose into three distinct types of metric Lie algebras.
Theorem I. Let g be a Lie algebra whose solvable radical r is abelian. Suppose that g is equipped with a nil-invariant symmetric bilinear form ·, · such that the metric radical g ⊥ of ·, · does not contain a non-trivial ideal of g. Let k × s be a Levi subalgebra of g, where k is of compact type and s has no simple factors of compact type. Then g is an orthogonal direct product of ideals where r = a × s * 1 and s = s 0 × s 1 are orthogonal direct products, and g 3 is a metric cotangent algebra. The restrictions of ·, · to g 2 and g 3 are invariant and non-degenerate. In particular, We split the proof into several lemmas. Consider the submodules of invariants r s , r k ⊆ r. Since s, k act reductively, we have Proof. The s-invariance of ·, · immediately implies a ⊥ b. Since r is abelian, r-invariance implies c ⊥ r. Since c ⊥ (s × k) by Corollary 5.5, this shows that c is an ideal contained in g ⊥ , hence c = 0. Now [k, r] ⊆ a and [s, r] = b by definition of a and b. Proof. The splitting as a direct product of ideals follows from Lemma 9.1. The orthogonality follows together with Corollary 5.5 and the fact that the s-invariance of ·, · implies s ⊥ a and k ⊥ b. Lemma 9.3. g ⊥ ⊆ k a and s b is a non-degenerate ideal of g.
Proof. z(g s ) = a, therefore g ⊥ ⊆ k a by Corollary C. Since also (s b) ⊥ (k a), we have To complete the proof of Theorem I, it remains to understand the structure of the ideal s b, which by Theorem A and the preceding lemmas is a Lie algebra with an invariant non-degenerate scalar product given by the restriction of ·, · . Proof. Since ·, · is r-invariant and r is abelian, b is totally isotropic. For the second claim, use b ∩ s ⊥ = 0 and the invariance of ·, · . Proof of Theorem I. The decomposition into the desired orthogonal ideals follows from Lemmas 9.2 to 9.5. The structure of the ideals g 2 and g 3 is Lemma 9.5.
The algebra g 1 in Theorem I is of Euclidean type. Let g = k V , with V ∼ = R n , be an algebra of Euclidean type and let k 0 be the kernel of the k-action on V . Proposition 6.13 and the fact that the solvable radical V is abelian immediately given the following: Proposition 9.6. Let g = k V be a Lie algebra of Euclidean type, and suppose that g is equipped with a symmetric bilinear form that is nil-invariant and g ⊥ -invariant, such that g ⊥ does not contain a non-trivial ideal of g. Then The following Examples 9.7 and 9.8 show that, in general, a metric Lie algebra of Euclidean type cannot be further decomposed into orthogonal direct sums of metric Lie algebras. Both examples will play a role in Section 10.
Example 9.7. Let k 1 = so 3 , V 1 = R 3 , V 0 = R 3 and g = (so 3 V 1 ) × V 0 with the natural action of so 3 on V 1 . Let ϕ : V 1 → V 0 be an isomorphism and put We can define a nil-invariant symmetric bilinear form on g by identifying V 1 ∼ = so * 3 and requiring . Then ·, · has signature (3,3,3) and metric radical h = g ⊥ , which is not an ideal in g. Note that ·, · is not invariant. Moreover, k 1 V 1 is not orthogonal to V 0 . A direct factor k 0 can be added to this example at liberty.
Example 9.8. We can modify the Lie algebra g from Example 9.7 by embedding the direct summand V 0 ∼ = R 3 in a torus subalgebra in a semisimple Lie algebra k 0 of compact type, say k 0 = so 6 , to obtain a Lie algebra f = (k 1 V 1 ) × k 0 . We obtain a nil-invariant symmetric bilinear form of signature (15,3,3) on f by extending ·, · by a definite form on a vector space complement of V 0 in k 0 . The metric radical of the new form is still g ⊥ = h.

Nil-invariant bilinear forms on Euclidean algebras
A Euclidean algebra is a Lie algebra e n = so n R n , where so n acts on R n by the natural action.
Example 9.9. Consider g = so 3 R n with a nil-invariant symmetric bilinear form ·, · , and assume that the action of so 3 is irreducible. By Proposition A.5, either so 3 ⊥ R n , or n = 3 and so 3 acts by its coadjoint representation on R 3 ∼ = so * 3 , and ·, · is the dual pairing. In the first case, R n is an ideal in g ⊥ , and in the second case, ·, · is invariant and non-degenerate.
Example 9.10. Let g be the Euclidean Lie algebra so 4 R 4 with a nil-invariant symmetric bilinear form ·, · . Since so 4 ∼ = so 3 × so 3 , and here both factors so 3 act irreducibly on R 4 , it follows from Example 9.9 that in g, R 4 is orthogonal to both factors so 3 , hence to all of so 4 . In particular, R 4 is an ideal contained in g ⊥ .
Theorem 9.11. Let ·, · be a nil-invariant symmetric bilinear form on the Euclidean Lie algebra so n R n for n 4. Then the ideal R n is contained in g ⊥ .
Proof. For n = 4, this is Example 9.10. So, assume n > 4. Consider an embedding of so 4 in so n such that R n = R 4 ⊕ R n−4 , where so 4 acts on R 4 in the standard way and trivially on R n−4 . By Example 9.10, so 4 ⊥ R 4 . Since R n−4 ⊆ [so n , R n ], the nil-invariance of ·, · implies so 4 ⊥ R n−4 . Hence, R n ⊥ so 4 .
The same reasoning shows that Ad(g)so 4 ⊥ R n , where g ∈ SO n . Then b = g∈SOn Ad(g)so 4 is orthogonal to R n . But b is clearly an ideal in so n , so b = so n by simplicity of so n for n > 4.
Theorem J. The Euclidean group E n = O n R n , n = 1, 3, does not have compact quotients with a pseudo-Riemannian metric such that E n acts isometrically and almost effectively.
Proof. For n > 3, such an action of E n would induce a nil-invariant symmetric bilinear form on the Lie algebra so n R n without non-trivial ideals in its metric radical, contradicting Theorem 9.11.
For n = 2, the Lie algebra e 2 is solvable, and hence by Baues and Globke [2], any nil-invariant symmetric bilinear form must be invariant. For such a form, the ideal R 2 of e 2 must be contained in e ⊥ 2 , and therefore, the action cannot be effective. Note that e 3 is an exception, as it is the metric cotangent algebra of so 3 .
Remark. Clearly, the Lie group E n admits compact quotient manifolds on which E n acts almost effectively. For example, take the quotient by a subgroup F Z n , where F ⊂ O n is a finite subgroup preserving Z n . Compact quotients with finite fundamental group also exist. For example, for any non-trivial homomorphism ϕ : R n → O n , the graph H of ϕ is a closed subgroup of E n isomorphic to R n , and the quotient M = E n /H is compact (and diffeomorphic to O n ). Since H contains no non-trivial normal subgroup of E n , the E n -action on M is effective. Theorem J tells us that none of these quotients admit E n -invariant pseudo-Riemannian metrics.

Simply connected compact homogeneous spaces with indefinite metric
Let M be a connected and simply connected pseudo-Riemannian homogeneous space of finite volume. Then we can write for a connected Lie group G acting almost effectively and by isometries on M , and H is a closed subgroup of G that contains no non-trivial connected normal subgroup of G (for example, G = Iso(M ) • ). Note that H is connected since M is simply connected. We decompose G = KSR, where K is a compact semisimple subgroup, S is a semisimple subgroup without compact factors and R is the solvable radical of G. We can therefore restrict ourselves in (10.1) to groups G = KR and connected uniform subgroups H of G.
The structure of a general compact homogeneous manifold with finite fundamental group is surveyed in Onishchik and Vinberg [14, II.5. § 2]. In our context, it follows that where V is a normal subgroup isomorphic to R n and L = KA is a maximal compact subgroup of G. The solvable radical is R = A V . Moreover, V L = 0. By a theorem of Montgomery [11] (also [14, p. 137]), K acts transitively on M . The existence of a G-invariant metric on M further restricts the structure of G.
Proof. Let Z(R) denote the center of R and N its nilradical. Since H is connected, It follows that Z(R) • is a connected uniform subgroup. Therefore, the nilradical N of R is N = T Z(R) • for some compact torus T . But a compact subgroup of N must be central in R, so T ⊆ Z(R). Hence, N ⊆ Z(R), which means R = N is abelian.
Combined with (10.2), we obtain Now assume further that the index of the metric on M is 2. Theorem D has strong consequences in the simply connected case.
Theorem H. The isometry group of any simply connected pseudo-Riemannian homogeneous manifold of finite volume and metric index 2 is compact.
Proof. We know from Theorem G that M is compact. Let G = Iso(M ) • , with G = KR as before. By Theorem D, R commutes with K and thus R = A by part 3 of Theorem G. It follows that G = KA is compact.
Then K is a characteristic subgroup of G which also acts transitively on M . Therefore, we may identify T x M at x ∈ M with k/(h ∩ k), where k is the Lie algebra of K. Hence, the isotropy representation of the stabilizer Iso(M ) x factorizes over a closed subgroup of the automorphism group of k. As this latter group is compact, the isotropy representation has compact closure in GL(T x M ). If follows that there exists a Riemannian metric on M that is preserved by Iso(M ). Hence, Iso(M ) is compact.
Remark. Note that, in fact, the isometry group of every compact analytic simply connected pseudo-Riemannian manifold has finitely many connected components (Gromov [6, Theorem 3.5.C]).
For indices higher than 2, the identity component of the isometry group of a simply connected M can be non-compact. This is demonstrated by the examples below in which we construct some M on which a non-compact group acts isometrically. The following Lemma 10.6 then ensures that these groups cannot be contained in any compact Lie group. Lemma 10.6. Assume that the action of K on V in the semidirect product G = K V is non-trivial. Then G cannot be immersed in a compact Lie group.
Proof. Suppose that there is a compact Lie group C that contains G as a subgroup. As the action of K on V is non-trivial, there exists an element v ∈ V ⊆ C such that Ad c (v) has nontrivial unipotent Jordan part. But by compactness of C, every Ad c (g), g ∈ C, is semisimple, a contradiction.
Example 10.7. Start with G 1 = ( SO 3 R 3 ) × T 3 , where SO 3 acts on R 3 by the coadjoint action, and let ϕ : R 3 → T 3 be a homomorphism with discrete kernel. Put The Lie algebras g 1 of G 1 and h of H are the corresponding Lie algebras from Example 9.7. We can extend the nil-invariant scalar product ·, · on g 1 from Example 9.7 to a left-invariant tensor on G 1 , and thus obtain a G 1 -invariant pseudo-Riemannian metric of signature (3,3) on the quotient M 1 = G 1 /H = SO 3 × T 3 . Here, M 1 is a non-simply connected manifold with a non-compact connected stabilizer.
In order to obtain a simply connected space, embed T 3 in a simply connected compact semisimple group K 0 , for example, K 0 = SO 6 , so that G 1 is embedded in G = ( SO 3 R 3 ) × K 0 . As in Example 9.8, we can extend ·, · from g 1 to g, and thus obtain a compact simply connected pseudo-Riemannian homogeneous manifold M = G/H = SO 3 × K 0 .
Example 10.8. Example 10.7 can be generalized by replacing SO 3 by any simply connected compact semisimple group K, acting by the coadjoint representation on R d , where d = dim K, ·, · for the adjoint representation is determined by its value H, H . Hence, it must be proportional to the Killing form.
A.2. Application to so 3 over the reals. Here, we consider the simple Lie algebra so 3 over the real numbers. Since so 3 has complexification sl 2 (C), we can apply Proposition A.4 to show: Proposition A.5. Let ·, · : so 3 × V → R be a skew pairing for the (non-trivial) irreducible module V . If the skew pairing is non-zero, then V is isomorphic to the adjoint representation of so 3 and ·, · is proportional to the Killing form.
Proof. Using the isomorphism of so 3 with the Lie algebra su 2 , we view so 3 as a subalgebra of sl 2 (C). We thus see that the irreducible complex representations of so 3 are precisely the su 2 -modules S k C 2 . Now let V be a real module for so 3 , which is irreducible and non-trivial, and assume that ·, · is a non-trivial skew pairing for V . We may extend V to a complex linear skew pairing ·, · C : sl 2 (C) × V C → C, where V C denotes complexification of the su 2 -module V .
In case V C is an irreducible module for sl 2 (C), Proposition A.4 shows that V C = V 2 is the adjoint representation of sl 2 (C). Hence, V must have been the adjoint representation of so 3 .
Otherwise, if V C is reducible, V is one of the modules V k = S 2 −1 C 2 with scalars restricted to the reals (cf. Bröcker and tom Dieck [4, Proposition 6.6]). It also follows that V C is isomorphic to a direct sum of S 2 −1 C 2 with itself. Since we assume that the skew pairing ·, · C for V C is non-trivial, Proposition A.4 implies that one of the irreducible summands of V C is isomorphic to S 2 C 2 . This is impossible, since k = 2 − 1 is odd.
The Killing form is always a non-degenerate pairing. In the light of the previous two propositions, this give us: Corollary A.6. Let ·, · : g × V → k be a skew pairing, where either g = sl 2 (k) or g = so 3 and k = R. Assume further that V g = {v ∈ V | gv = 0} = 0. Define Then either V ⊥ = 0 or V ⊥ = g.
Proof. The first case occurs precisely if there exists an irreducible summand W of V on which the restricted skew pairing g × W → k induces the Killing form.

Appendix B. Nil-invariant scalar products on solvable Lie algebras
We present a new proof for a key result of Baues and Globke [2, Theorem 1.2]. The importance of this result lies in it being the crucial ingredient in the proof of our Theorem A, which supersedes it and is itself the fundamental tool in the study of Lie algebras with nil-invariant bilinear forms.
We recall some well-known facts (see Jacobson [8,Chapter III]). Let g be an arbitrary finitedimensional real Lie algebra. For X ∈ g, let g(X, 0) denote the maximal subspace of g on which ad(X) is nilpotent. Let H 0 be a regular element of g, that is, dim g(H 0 , 0) = min{dim g(X, 0) | X ∈ g}. We write g 0 = g(H 0 , 0) for short. Then, by [8, Chapter III, Theorem A, Proposition 1.1], g 0 is a Cartan subalgebra of g, and there is a Fitting decomposition g = g 0 ⊕ g 1 into g 0 -submodules. In particular, as a Cartan subalgebra, g 0 is nilpotent, and the restriction of ad(H 0 ) to g 1 is an isomorphism.
Proof. The set of regular elements in g is Zariski-open, and thus intersects g 0 in a nonempty Zariski-open set (it contains H 0 ). So, any X ∈ g 0 sufficiently close to H 0 is also a regular element. Two Cartan subalgebras with a common regular element coincide [8, p. 60], so that g 0 = g(X, 0). Lemma B.3. Let h be any nilpotent subalgebra of g. Then the restriction of ·, · to h is an invariant bilinear form on h.
Proof. Let H ∈ h. By nil-invariance of ·, · , the nilpotent part ad g (H) n of the Jordan decomposition of ad g (H) is skew-symmetric with respect to ·, · . Since h is a nilpotent subalgebra, ad h (H) is a nilpotent operator, and hence ad g (H) n | h = ad h (H). This means the restriction of ·, · to h is an invariant bilinear form.
Proof of Theorem B.1. Suppose that g is solvable. Let H 0 be a regular element in g. Then g 1 is contained in the nilradical n of g. Indeed, n ⊇ [g, g] and g 1 = ad(H 0 )g 1 ⊆ [g, g].
Suppose now that g has a nil-invariant symmetric bilinear form ·, · . In particular, ad(N ) is skew-symmetric for all N ∈ n and the restriction of ·, · to any nilpotent subalgebra is invariant by Lemma B.3. In particular, the restriction of ·, · to the Cartan subalgebra g 0 is invariant. Now let X ∈ g. Then, for any N, N ∈ n, Thus, ad(X) is skew-symmetric for the restriction of ·, · to RX + n, and moreover X ⊥ [X, n].
Observe that g 1 ⊆ [H 0 , g 1 ] ⊆ [H 0 , n], and hence H 0 ⊥ g 1 . The same holds for all elements X in a non-empty open subset of g 0 (compare Lemma B.2), and hence g 0 ⊥ g 1 .
Altogether, any X ∈ g 0 preserves ·, · on g 1 , and, as stated before, preserves ·, · on g 0 , since g 0 is nilpotent. Hence, ad(X) is skew-symmetric on g. Since g = g 0 + n, this means ·, · is an invariant bilinear form on g.