Is finite dimensional central division algebra = $\otimes$(proper central division subalgebra)

Question

One exercise in Jacobson's Basic Algebra II is approximately the following.

If $\Delta_1,\Delta_2$ are finite dimensional central division algebra over $F$, and if $\operatorname{gcd}([\Delta_1:F],[\Delta_2:F])=1$, then $\Delta_1\otimes_F\Delta_2$ is a central division algebra.

I can do this, and I am now courious about the converse problem.
That is,

given a finite dimensional central division algebra $\Delta$, do there always exist central division algebras $\Delta_1,\Delta_2$ with coprime dimensions such that $\Delta\cong \Delta_1\otimes_F\Delta_2$?

Of course, to exclude trivial answers, we require that $\Delta_1,\Delta_2\neq F$ and that $[\Delta:F]$ is not a prime power.

I guess that a possible way is to find a central subalgebra $\Delta_1$ of $\Delta$ whose dimension $[\Delta_1:F] $ is coprime to $[\Delta:\Delta_1]$.
Once this is done, a theorem in the book says that $\Delta=\Delta_1\otimes_F C_{\Delta}(\Delta_1)$.
Here $C_\Delta(\Delta_1)$ is the centralizer of $\Delta_1$ in $\Delta$, which will be division and whose center will (by that theorem) be the center of $\Delta$, namely, $F$.
Also, $[C_\Delta(\Delta_1):F]=[\Delta:F]/[\Delta_1:F]=[\Delta:\Delta_1]$, which is coprime to $[\Delta_1:F]$ by construction.
These give what I want.

Answer 1

Yes, there is such a "primary decomposition" type result for central division algebras, see Gille, Szamuely, Central Simple Algebras and Galois Cohomology, Proposition 4.5.16 (attributed to Brauer):

Let $k$ be a field, let $D$ be a central division $k$-algebra of index $\mathrm{ind}(D) = p_{1}^{m_{1}} \dotsb p_{r}^{m_{r}}$. Then there exist central division $k$-algebras $D_{1},\dotsc,D_{r}$ such that $\mathrm{ind}(D_{i}) = p_{i}^{m_{i}}$ and $D \simeq D_{1} \otimes_{k} \dotsb \otimes_{k} D_{r}$.