$\DeclareMathOperator\SL{SL} \newcommand{\gl}{\mathfrak{gl}} \newcommand{\sl}{\mathfrak{sl}}$One of my graduate students asked me for a reference for the following fact. Let $k$ be a general field (she's particularly interested in $k = \mathbb{F}_2$) and let $n \geq 2$. Consider the representation $\gl_n(k)$ of $\SL_n(k)$. Let $V$ be a nonzero proper subrepresentation of $\gl_n(k)$. Then $V$ is either

1. the $1$-dimensional line of scalar matrices; or
2. the subspace $\sl_n(k)$ of trace-$0$ matrices.

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

(thanks to YCor for cleaning up my original statement)