Let $B$ be a finite-dimensional $F$-algebra. The Jacobson radical of $B$ is the intersection of all maximal left ideals of $B$.
The following is an exercise from Voight's Quaternion Algebras.
Suppose the characteristic of $F$ is not equal to $2$. Compute the Jacobson radical of $B$ with basis $1, i, j, ij$ satisfying $i^2=a, j^2=0,$ and $ij=-ji.$
I'm trying to compute the Jacobson radical of quaternion orders given an explicit basis and this seems like a good exercise to get an idea of how to compute the Jacobson radical. Can anyone help?