When trying to do Lie-theoretic calculations on Lie groups (finding the Bruhat decomposition, etc.) I've often come across expositions that seem to be implicitly using a result something like the following:
Let $G$ be a Lie group and $H < G$ be a subgroup such that $H = G^\sigma$ (the points of $G$ fixed by $\sigma$), where $\sigma$ is an involution of $G$. Let $T$ be a maximal torus, and $B$ be a Borel subgroup containing $T$, such that $B$ and $T$ and $\sigma$-stable. Then $T^\sigma$ and $B^\sigma$ are a maximal torus and Borel subgroup of $H$, respectively.
For instance, something like this is done in Lakshmibai and Billey, Singular Loci of Schubert Varieties.
Question: What is the actual statement of this result in its full generality? What is the proof, and/or where is it written down?
Thanks in advance.