$P_-$ of ${\EuScript{J}}$-negative vectors; we have $p+q=\dim V$ and $q$ is known as the *index* of ${\EuScript{J}}$.

4.  For every linear map $L:V\to{{\mathbb R}}$ there exists a unique $v\in V$ such that $L(w)=(v,w)$ for each $w\in V$.

5.  For each $L:V\to V$ linear there exists a unique linear operator $L^+:V\to V$ (the pseudo-adjoint) such that $(L(v),w)=(v,L^+(w))$ for every $v,w\in V$.

6.  Every pseudo-self-adjoint $L:V\to V$, that is, such that $L=L^+$, satisfies

    1.  eigenspaces corresponding to distinct eigenvalues are pseudo-orthogonal;

    2.  if a subspace $E$ is $L$-invariant, then $E^\perp$ is also $L$-invariant.

The proofs are rather standard and can be found in [@Maltsev63].

The following simple result will be very useful in what follows.

\[le:kuhne\] Let $V$ be a real finite dimensional vector space endowed with a non-positive definite and non-degenerate quadratic form ${\EuScript{J}}:V\to{{\mathbb R}}$.

If a symmetric bilinear form $F:V\times V\to{{\mathbb R}}$ is non-negative on $C_0$ then $$\begin{aligned}
  r_+=\inf_{v\in C_+} \frac{F(v,v)}{\langle Jv,v\rangle}
  \ge \sup_{u\in C_-}\frac{F(u,u)}{\langle Ju,u\rangle}=r_-\end{aligned}$$ and for every $r$ in $[r_-,r_+]$ we have $F(v,v)\ge r\langle Jv,v\rangle$ for each vector $v$.

In addition, if $F(\cdot,\cdot)$ is positive on $C_0\setminus\{0\}$, then $r_-<r_+$ and $F(v,v)>
r\langle Jv,v\rangle$ for all vectors $v$ and $r\in(r_-,r_+)$.

\[rmk:Jseparated\] Lemma \[le:kuhne\] shows that if $F(v,w)=\langle \tilde J
  