the intersection of a real form and a parabolic subalgebra

Question

Let $\mathfrak g$ be a complex semisimple Lie algebra, $\mathfrak g_0$ a real form and $\tau$ the conjugation of $\mathfrak g$ with respect to $\mathfrak g_0$. Real Lie algebra $\mathfrak g_{0}$ is called a real form of a complex Lie algebra $\mathfrak g$ if $\mathfrak g$ is the complexification of $\mathfrak g_{0}$. Now we fix a parabolic subalgebra $\mathfrak p$ of $\mathfrak g$. I want to show that $\mathfrak g_{0} \cap \mathfrak p$ is a real form of $\mathfrak p \cap \tau\mathfrak p$. I'm completely stuck on how to start this question. Any help and comments will be appreciated.

Answer 1

For any $A \in \mathfrak{p} \cap \tau(\mathfrak{p})$ one has $$2 A=(A+\tau(A))+(A-\tau(A)),$$ where the first summand is contained in $\mathfrak{p} \cap \mathfrak{g}_{0}$ and the second one gets into that subspace after multiplication by $i$. In fact, this argument is valid for any complex Lie subalgebra $\mathfrak{p} \subset \mathfrak{g}$.