Exercise 11, § 13

The notations are those of no. 4.VII, type $D_l(l\geq$ 3). Show that $\mathbf{O}^+_0(\Psi )$ is the commutator subgroup of $\mathbf{S}\mathbf{O}(\Psi )$. (Apply Algebra, Chap. IX, §6, Exerc. $17b$) and Algebra, Chap. IX, §9, Exerc. $11b).)$ Deduce that Aut$_e(\mathfrak{g})$ is equal to the image of $\mathbf{O}^+_0(\Psi )$ in $\mathbf{S}\mathbf{O}(\Psi )/\{\pm 1\}$. Moreover, $-1$ belongs to $\mathbf{O}^+_0(\Psi )$ if and only if $l$ is even or $-1$ is a square in $k($Algebra, Chap. IX, §9, Exerc. $11c))$.