Maximal central subalgebra(s) in non-central division $k$-algebra

Question

Let $D$ be a finite dimensional (non-central) division $k$-algebra, where $k$ is a field. Is there a concrete description of the maximal subalgebra(s) of D having center $k$?

Answer 1

Here is a partial answer.

Every subalgebra of $D$ is division. So an essentially tautological answer is that the subalgebras of $D$ which are central over $k$ are the central division algebras $A/k$ which embed in $D$.

If you are working in certain settings, e.g., $k$ is a number field, this may be useful to reduce your problem to a local embedding problem.

If we restrict to the case where $D$ is a quaternion algebra, there is a simple answer. (Maybe assume char $k \ne 2$ to be safe, though you might not have too.)

Note the center of $D$ is a field $K$, and necessarily a finite-degree extension of $k$. If $A$ is a non-commutative subalgebra of $D$, then it must be quaternion over its center $F$, and $D \simeq A \otimes_F K$. So the only case where $D$ has a non-trivial $k$-central subalgebra, is when $D \simeq A \otimes_k K$ for a $k$-central quaternion algebra $A$, and then the subalgebra is just $A$ (up to isomorphism).

The same argument applies if $D$ has prime degree over its center (possibly assuming char $k \ne$ degree).

If $D$ does not have prime degree over its center, then there are more possiblities for subalgebras so things will get a little more complicated but I think a similar argument can be used.