Suppose $D$ is a central division algebra over $\mathbb{Q}$ of degree $n$. Let $A \subset D$ be a subalgebra, say also central over $\mathbb{Q}$ (to restrict the degree of generality). Now if a field $L$ (e.g. $L= \mathbb{R}$) splits $D$, then what can we say about about the splitting behaviour of $A$ with respect to $L$?
If $L$ can both split $A$ or not, do we have easy concrete examples? To avoid trivial situations, suppose that $\deg A \geq 2$ and that $A \neq D$.
Since $A$ is supposed to be central, the answer is YES.
Since $A$ is central and simple (thanks, Jyrki), the Centralizer theorem shows that $D\simeq A\otimes_k B$, where $B=C_D(A)$
I claim that $A,B$ have coprime degrees. Since we work over $\mathbb{Q}$, and since $D,A,B$ are division, $\deg(A)\deg(B)=\deg(D)=\mathrm{ind}(D)=\exp(D)=\mathrm{lcm}(\exp(A),\exp(B))=\mathrm{lcm}(\deg(A), \deg(B))$. Hence $\gcd(\deg(A),\deg(B))=1.$
Now if $D_L$ is split (where $D_L=D\otimes_k L)$, then $A_L\sim B^{op}_L$ (where $\sim$ is the Brauer equivalence), so $\mathrm{ind}(A_L)=\mathrm{ind}(B^{op}_L)=\mathrm{ind}(B_L)$
Now $\mathrm{ind}(A_L)\mid\deg(A_L)=\deg(A)$, and similarly $\mathrm{ind}(B_L)\mid \deg(B)$. Hence $\mathrm{ind}(A_L)=\mathrm{ind}(B_L) $ divides $\gcd(\deg(A),\deg(B))=1.$ Hence $A_L$ is split and $B_L$ is split.