Exercise 7, § 2

Assume that $k$ is an ordered field. Let $(\mathfrak{g},\mathfrak{h})$ be a split semi-simple Lie algebra of rank $l$; put dim$\mathfrak{g}=l+ 2m$.

a) Show that the Killing form $\Phi$ of $\mathfrak{g}$ is the direct sum of a neutral form of rank $2m$ and a positive non-degenerate form of rank $l$; in particular, its index is $m$.

b) Let $\varphi$ be an involutive automorphism of $\mathfrak{g}$ whose restriction to $\mathfrak{h}$ is $-$Id. Show that the form

$^{10}$For an explicit calculation of this discriminant, see: T. A. SPRINGER and R.

STEINBERG, Conjugacy Classes (no. 4.8), Seminar on Algebraic Groups and

Related Finite Groups, Lect. Notes in Math. 131, Springer-Verlag (1970).

is symmetric, non-degenerate, and positive.

c) Let $k=k(\overline{\alpha})$, where $\alpha$ is an element $<0$ in $k$. Denote by $c$ the non-trivial $k$-automorphism of $k'$. Let $\mathfrak{g}_c$ be the $k$-subspace of $\mathfrak{g}_{(k')}=k'\otimes_k\mathfrak{g}$ formed by the elements $y$ such that

Show that $\mathfrak{g}_c$ is a $k$-Lie subalgebra of $\mathfrak{g}_{(k')}$ and that the injection of $\mathfrak{g}_c$ into $\mathfrak{g}_{(k')}$ extends to an isomorphism from$\surd k'\otimes_k\mathfrak{g}_c$ to $\mathfrak{g}_{(k')}$. The algebra $\mathfrak{g}_c$ is semi-simple, and $\alpha \mathfrak{h}$ is a Cartan subalgebra of it.

d) Show that the Killing form of $\mathfrak{g}_c$ is negative. Deduce that $\mathfrak{g}_c$ is not splittable (unless $\mathfrak{g}= 0$).

e) When $k=\mathbf{R}$, show that Int($\mathfrak{g}_c$) is compact.