Underlying real Lie algebra

Core idea

For a complex Lie algebra $\mathfrak g$ , the underlying real Lie algebra $\mathfrak g_{\mathbb R}$ is obtained by realifying its vector space: it has the same additive group and bracket, but scalar multiplication is restricted along $\mathbb R\hookrightarrow\mathbb C$ . If $\dim_{\mathbb C}\mathfrak g=n$ , then

\dim_{\mathbb R}\mathfrak g_{\mathbb R}=2n.

Complex-linear Lie algebra homomorphisms become real-linear homomorphisms, so $(-)_{\mathbb R}$ is a functor.

Complex structure retained as extra data

Multiplication by $i$ defines a real-linear endomorphism $J$ of $\mathfrak g_{\mathbb R}$ satisfying $J^2=-1$ and

[JX,Y]=J[X,Y]=[X,JY].

Forgetting $J$ can lose information: an isomorphism of the underlying real Lie algebras need not be complex-linear.

This scalar restriction is not the inverse of complexification. In fact,

\mathfrak g_{\mathbb R}\otimes_{\mathbb R}\mathbb C \cong \mathfrak g\oplus\overline{\mathfrak g}

as complex Lie algebras.

References

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhäuser, 2002, Chapter I. Publisher record.
Nathan Jacobson, Lie Algebras, Dover, 1979, Chapter I.