Algebra isomorphisms

Question

Let $A_1,A_2,B\subset A$ be (if necessary finite dimensional) $k$-algebras (if necessary $k$ can be assumed to be a field) with $A = A_1\oplus A_2$ and $A = A_1 + B$. One may additionally assume that $B\subset A$ have the same $1$. Is it always true that $A_2$ is isomorphic to a subalgebra of $B$? (Even with the same $1$ as $B$?)

Answer 1

No, $A_2$ is isomorphic to a factor: $$ A_2\cong A/A_1 =(A_1+B)/A_1\cong B/(A_1\cap B). $$