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**topspin1617** Let $\displaystyle k$ be a field. Let $\displaystyle A$ be the $\displaystyle k$-subspace of $\displaystyle M_3(k)$ spanned by

$\displaystyle 1=\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}, \alpha =\begin{pmatrix}0&1&0\\1&0&0\\0&0&-1\end{pmatrix},\beta =\begin{pmatrix}0&0&0\\0&0&0\\1&1&0\end{pmatrix}$,

which can be shown to actually be a $\displaystyle k$-algebra.

Determine the Jacobson radical of $\displaystyle A$, which is the intersection of all maximal left ideals.

I know at least $\displaystyle \beta$ (and hence the left ideal it generates) is in the Jacobson radical, because it is nilpotent (it is known that the Jacobson radical contains all nilpotent elements). I don't know where to proceed from here, though...