I need a reference for the following statement:


Let $G$ be a linear algebraic group over algebraically closed field $k.$ Let $V$ be a finite dimensional $G$-module. Then $V$ is subrepresentation of $k[G]^n$ for some $n$ where $k[G]$ is coordinate ring of $G.$


I could find this statement in Steinberg's lecture notes on "Conjugacy Classes in Algebraic groups" but am not happy with the proof there.


Thanks in advance.



Any representation is a subrepresentation of a direct sum of the regular representation
This is the unique Lemma in section 3.5 of Waterhouse's "Introduction to Affine Group Schemes". It only requires that $G$ be an affine group scheme over a field.


