Cartan subalgebra

Cartan subalgebra

Let $g$ be a real semisimple Lie algebra with Cartan decoposition $(l,p)$.
How can we show that a Cartan subspace $a$ of $p$ (Cartan subspace of $p
=$ maximal element in a set that consists of all Lie subalgebras of $g$
that are in $p$) iff $a_{\mathbf C} = a + ia$ (here $a_{\mathbf C}$ is the
complexification of $a$) is a Cartan subspace of $p_{\mathbf C}$.
And we know that every Cartan subspace of $p$ is a commutative Lie
subalgebra of $p$.