(Algebras are associative with unity in this question. They are not necessarily commutative.)
Given an $\mathbb{R}$-subalgebra $S$ of an algebra $A$, write $C(S)$ for the $\mathbb{R}$-algebra of all $t \in A$ such that for all $s \in S$ we have $st = ts$. We always have $C(C(A)) \supseteq A$ as a consequence of general facts about Galois connections.
Question. Which subalgebras $S$ of the matrix algebra $M_n(\mathbb{R})$ satisfy $C(C(S)) = S$?
Idea. I was thinking of looking at linear subspaces of $\mathbb{R}^n$ and decomposing each element of $\mathbb{R}^n$ as a sum of something in that subspace and something in its orthogonal complement. Algebras of matrices that act on just one of these components might have the desired property (but I'm not sure). I also don't know if this is an exhaustive list.
In theory, this result should be helpful, but I can't understand what it's saying.