Let $G \subseteq GL(n,\mathbb{R})$ be a subgroup. I want to consider the set $$\tilde G = \{a \in \mathbb{R}^{n \times n}: \; a^T G a = G\}$$ of matrices that preserve $G$ by conjugation by transpose.
This is a subgroup of $GL(n,\mathbb{R})$, because: if $a \in \tilde G$ then $a^T I a \in G$ is invertible and therefore $\mathrm{det}(a) \ne 0$; also, if $a,b \in \tilde G$ then $$(ab)^T G ab = b^T a^T G ab = b^T G b = G$$ and $$(a^{-1})^T G a^{-1} = (a^{-1})^T a^T G a a^{-1} = G.$$
Question: (1) Is there a common name for the operation $G \mapsto \tilde G$?
(2) What does the operation do to Lie algebras? From the examples of $O(n,\mathbb{R})$ and $SP(2n,\mathbb{R})$ I would guess that if $\mathfrak{g}$ is the Lie algebra of $G$ then $$\mathfrak{\tilde g} = \{a \in \mathbb{R}^{n \times n}: a^T \mathfrak{g} + \mathfrak{g} a = 0\}$$ but on second thought I'm not even sure this is a Lie algebra at all.