Real form

Remarks

Equivalently, the $\C$-linear map $\mathfrak{g}_0 \otimes_\R \C \to \mathfrak{g}$, $X \otimes z \mapsto zX$, is an isomorphism of complex Lie algebras. Equivalently again, $\mathfrak{g}_0$ is the set of fixed points of a conjugation of $\mathfrak{g}$: a map $\sigma$ that is additive, satisfies $\sigma(zX) = \bar z \sigma(X)$, preserves brackets and has $\sigma^2 = \id$. Example: $\mathfrak{sl}_2(\R)$ is a real form of $\mathfrak{sl}_2(\C)$; it is closed under brackets, and each complex matrix of trace $0$ is uniquely $X + iY$ with $X, Y$ real of trace $0$. Non-example: $i\,\mathfrak{sl}_2(\R)$ satisfies the direct sum condition but is not a subalgebra, since $[ih, ie] = -[h, e] = -2e$ is real, where $h = \operatorname{diag}(1, -1)$ and $e = E_{12}$. Real forms of Lie groups, see real form of a Lie group, have real forms of Lie algebras as their Lie algebras.