For small values of $l$, isomorphisms occur among certain of the classical algebras. Show that $B_2$ is isomorphic to $C_2$.
Well, both $B_2$ and $C_2$ have dimension $10$. $B_2$ consists of $5\times 5$ matrices, while $C_2$ consists of $4\times 4$ matrices. It seems to take too much work to exhibit an explicit isomorphism, and then computing to show that $\phi([x,y])=[\phi(x),\phi(y)]$, where $[x,y]=xy-yx$. How can we show that the two algebras are isomorphic?