I'm looking for a reference for the following standard result: 


Let $U$ be a unipotent algebraic group over an algebraically closed field $k$ (of any characteristic); then any algebraic representation of $U$ has a fixed point. 


Statements of Engel's theorem for the analogous statement about Lie algebras seem to be ubiquitous. I can also find the statement that connected solvable groups always preserve a line in many places (for example, Borel, Theorem III.10.4). Combining this with the fact that unipotent groups have no non-trivial characters gives me the result I need. But it would be nice to have a place to which I could refer for the precise statement about unipotent groups. 



Reference request: representations of unipotent groups have a fixed point.
Theorem 17.5 in Humphreys's *Linear Algebraic Groups* seems to be the result you want. (Also, doesn't the result for solvable groups only imply the corresponding result for *connected* unipotent groups? The proof for unipotent groups doesn't require connectedness.)


