Exercise 7, § 5

Assume that $\mathfrak{g}$ is splittable. Let $B(\mathfrak{g})$ be the canonical basis of the canonical Cartan subalgebra of $\mathfrak{g}$ (no. 3, Remark 2). If $s\in$ Aut($\mathfrak{g}$)$,s$ induces a permutation of $B(\mathfrak{g})$; denote by sgn($s$) the sign of this permutation. Show that $s$ operates on $\bigwedge^n\mathfrak{g}$ (with $n=$ dim $\mathfrak{g}$) by

$x \rightarrow$ sgn($s$)$.x$.