# Complexification of a Lie subalgebra of a compact real form

I'm currently reading the paper Lie algebra Cohomology and the Generalized Borel–Weil theorem written by B. Kostant, and I have a question about Remark 3.9 he made.

In this paper, $$\mathfrak{g}$$ is a complex semisimple Lie algebra and $$\mathfrak{t}$$ is its compact real form. Using the decomposition $$\mathfrak{g}=\mathfrak{t}\oplus i\mathfrak{t}$$ the author defines the complex conjugate $$\ast$$ on $$\mathfrak{g}$$ by defining $$(a+ib)^{\ast}=-a+ib$$ for $$a,b\in\mathfrak{t}$$. (He uses the subspace $$i\mathfrak{t}$$ as a real part.) In Remark 3.9 of the paper, the author mentions that

In case $$\mathfrak{a}$$ is a Lie subalgebra and $$\mathfrak{a}=\mathfrak{a}^{\ast}$$ it follows that $$\mathfrak{a}$$ is necessarily reductive in $$\mathfrak{g}$$ since it clearly arises as the complexification of a Lie subalgebra of $$\mathfrak{t}$$.

First of all, it was not clear to me what does it mean by saying that $$\mathfrak{a}$$ is reductive ‘in’ $$\mathfrak{g}$$. Does it simply mean that $$\mathfrak{a}$$ is a reductive Lie algebra? Secondly, if it means that $$\mathfrak{a}$$ is a reductive Lie algebra, then I am curious why the fact that $$\mathfrak{a}$$ arises as a complexification of a Lie subalgebra of $$\mathfrak{t}$$ implies that $$\mathfrak{a}$$ is reductive.

Historically, I think the paper may simply be early enough that it wasn't clear whether the various notions of reductivity in characteristic $$0$$ were equivalent. “Reductive in $$\mathfrak g$$” could mean, for example, that the adjoint representation on $$\mathfrak g$$ was completely reducible.
A Lie subalgebra of $$\mathfrak k$$ is the Lie algebra of a compact group, so its complexification is the Lie algebra of the complexification of a compact group, which is reductive.
(By the way, the letter is a Fraktur k, $$\mathfrak k$$ \mathfrak k, not a Fraktur t, $$\mathfrak t$$ \mathfrak t (setting aside the usual issues of Fraktur vs. Sütterlin, for which you can read elsewhere on MO (e.g.)). This matters to the extent that it is very hard for a Lie theorist, or at least for me, to read $$\mathfrak t$$ as being anything other than the Lie algebra of a torus.)