Suppose $D$ is an explicitly given rank 9 central simple algebra over ${\mathbb Q}$ (or a number field). For example it could be specified by two cubic subfields ${\mathbb Q}(a), {\mathbb Q}(b)\subset D$ and one relation $ba=\sum a_i b_j$. What is the best way to determine whether $D$ is a matrix algebra or a division algebra?

Any ideas or references would be highly appreciated! I simply tried to look for random elements whose minimal polynomial is reducible (sort of like Parker's meataxe for group representations), but this is not terribly efficient, and I don't know how to do this cleverly.

Suppose $D$ is an explicitly given rank 9 central simple algebra over ${\mathbb Q}$ (or a number field). For example it could be specified by two cubic subfields ${\mathbb Q}(a), {\mathbb Q}(b)\subset D$ and one relation $ba=\sum a_i b_j$. What is the best way to determine whether $D$ is a matrix algebra or a division algebra?

Any ideas or references would be highly appreciated! I simply tried to look for random elements whose minimal polynomial is reducible (sort of like Parker's meataxe for group representations), but this is not terribly efficient, and I don't know how to do this cleverly. Perhaps one could also take many cubic subfields and record which primes split in them - if the union of split primes over all subfields seems to cover all prime numbers, this suggests that $D$ is a matrix algebra; again, I don't know how to make this into an actual algorithm.