# Wednesday September 11

**Theorem:**
If $L$ is semisimple and $x\in L$, there exists a unique $x_s, x_n$ in $L$ such that $x = x_s + x_n$, $[x_n, x_s] = 0$, $\ad x_s$ is semisimple, and $\ad x_n$ is nilpotent.

> Todo.