Let $(A,\cdot)$ be a finte dimensional associative algebra over $\mathbb{C}$, which is noncommutative, and $(\mathfrak{g},[\cdot,\cdot])$ be its induced Lie algebra, i.e., $\mathfrak{g}= A$ as vector spaces and $[x,y]:=x\cdot y-y\cdot x,\forall x,y\in{A}.$ If
- the annihilator $ann(A):=\{x\in A:x\cdot y=y\cdot x=0,\forall y\in A\}$ is zero. (Note that there do exist associative algebras having non-trivial annihilator, for exmaple, nilpotent associative algebras).
- $(\mathfrak{g},[\cdot,\cdot])$ is a reductive Lie algebra, i.e., the adjoint representation ad$:\mathfrak{g}\rightarrow\mathfrak{gl}(\mathfrak{g})$ is completely reducible.
Then is $(A,\cdot)$ necessarily a semisimple associative algebra? If not, are there any counterexamples?