习题 11, § 13

记号是no. 4.VII、型 $D_l(l\geq$ 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 $\mathbf{S}\mathbf{O}(\Psi )/\{\pm 1\}$. 此外，$-1$属于到$\mathbf{O}^+_0(\Psi )$当且仅当$l$是偶数或$-1$是$k($代数中的平方，第1章。 IX, §9，锻炼。 $11c))$。