Exercise 9, § 13

The notations are those of no. 3.VII, type $C_l(l\geq 1)$. In particular, we identify Aut$_e(\mathfrak{g})$ with a subgroup of $\mathbf{S}\mathbf{p}(\Psi )/\{\pm 1\}$. Show that the image in $\mathbf{S}\mathbf{p}(\Psi )/\{\pm 1\}$ of a symplectic transvection (Algebra, Chap. IX, §4, Exerc. 6) belongs to Aut$_e(\mathfrak{g})$. Deduce that Aut$_e(\mathfrak{g}) =\mathbf{S}\mathbf{p}(\Psi )/\{\pm 1\}($Algebra, Chap. IX, §5, Exerc. 11), and that Aut($\mathfrak{g}$)$/$Aut$_e(\mathfrak{g})$ can be identified with $k^*/k^{*2}$.