$\DeclareMathOperator\SL{SL}$
Let $V$ be the adjoint representation of $\SL_n(\mathbb{F}_2)$, so $V$ is the space of trace zero $n \times n$ matrices over $\mathbb{F}_2$.  For a paper she is writing, one of my graduate students asked me for a reference for the following fact:

1. if $n$ is odd, then $V$ is irreducibe; and
2. if $n$ is even, then the only nontrivial subrepresentation of $V$ is the subspace $W \cong \mathbb{F}_2$ of scalar matrices.

I didn't know one off the top of my head.  Can anyone provide such a reference?