Let be a field. Let be the -subspace of spanned by

,

which can be shown to actually be a -algebra.

Determine the Jacobson radical of , which is the intersection of all maximal left ideals.

I know at least (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...