Compact real form

Remarks

Example: $\mathfrak{su}(n)$, the skew-Hermitian $n \times n$ matrices of trace $0$, is a compact real form of $\mathfrak{sl}_n(\C)$. Its Killing form is $B(X, Y) = 2n \tr(XY)$, and for $X$ skew-Hermitian and nonzero, $\tr(X^2) = -\tr(X X^*) < 0$. Non-example: $\mathfrak{sl}_2(\R)$ is a real form of $\mathfrak{sl}_2(\C)$ but not a compact one, since $h = \operatorname{diag}(1, -1)$ has $B(h, h) = 8 > 0$. A real semisimple Lie algebra has negative definite Killing form if and only if it is the Lie algebra of a compact Lie group. Every complex semisimple Lie algebra has a compact real form, and any two are conjugate under an automorphism of $\mathfrak{g}$.