Exercise 7, § 13

The notations are those of no. 2.VII, type $B_l$. Show that $\mathbf{O}^+_0(\Psi )$ is the commutator subgroup of $\mathbf{S}\mathbf{O}(\Psi )$. (Remark that $\mathbf{O}(\Psi )$ is equal to $\{\pm 1\} \times \mathbf{S}\mathbf{O}(\Psi )$, hence has the same commutator subgroup as $\mathbf{S}\mathbf{O}(\Psi )$, and apply Algebra, Chap. IX, §9, Exerc. $11b$)$.)$ Deduce that Aut$_e(\mathfrak{g}) =\mathbf{O}^+_0(\Psi )$.