Exercise 6, § 1

Show that a (real) Lie algebra is compact if and only if it admits a basis $(e_{\lambda})_{\lambda\in L}$ for which the structure constants $\gamma_{\lambda \mu\nu}$ are anti-symmetric in $\lambda , \mu, \nu$ (that is, they satisfy $\gamma_{\lambda \mu\nu}=-\gamma_{\mu\lambda \nu}=-\gamma_{\lambda \nu \mu}$).