According to the Wikipedia article on complex Lie algebras , any complex Lie algebra $\mathfrak{g}$ that is isomorphic to its conjugate $\overline{\mathfrak{g}}$ admits a real form.
So as I understand it, this isomorphism exists if and only if there is an antilinear automorphism $\tau: \mathfrak{g} \to \mathfrak{g}$. The wikipedia article then says 'we may assume without loss of generality that $\tau$ is the identity on the underlying real vector space', which seems nonsensical and wrong to me, as this would imply that tau is the identity and hence not antilinear. Anyway, the rest of the argument depends on $\tau$ being an involution; I believe that part is correct, so the question is, why can we assume that $\tau$ is an involution?
My idea was basically that $\tau^2$ is a complex linear automorphism; if $\tau$ is diagonalizable, then $\tau^2=\varphi^2$ for a linear isomorphism $\varphi$. However, I do not know whether this $\varphi$ need be a Lie algebra automorphism; I suspect not. If it is, however, then $\tau \circ \varphi^{-1}$ is the antilinear involution we are looking for. Either way, this depends on $\tau^2$ being diagonalizable, so I do not have much hope for this argument.
Does anyone know a proof (or counterexample) of this fact? For reference, I am currently reading Lie groups, Representation theory and Symmetric spaces due to Ziller, where I have finished the first two chapters.
