I know this is not a research level question, but may be I can get an idea...

Is there a direct proof of the following without going through composition series or Artin-Wedderburn theorem?

Let $V$ be a finite-dimensional complex vector space. Let $A \subset End(V)$ be a self-adjoint subalgebra. Then $A$ is semisimple. I am using the following definition of semisimple algebra: semisimple algebra is a direct sum of simple algebras, and a simple algebra is one with no two-sided ideals other than $\{0\}$ and itself. thanks!