Why are algebraic groups over a general field defined to be algebraic varieties?

Question

In definitions of algebraic varieties over a field $k$, sources always seem to assume $k$ is algebraically closed. On the other hand, many uses of algebraic groups (for number theory at least) require us to be over a more general $k$, and yet they always seem to be defined as a certain kind of algebraic variety. Are authors implicitly considering the algebraic group as over $\bar{k}$ instead? Or are they implicitly taking a more general definition of variety? Or have I just not checked and correctly remembered enough sources?