I am reading Maxwell Rosenlicht's 1956 paper, "Some Basic Theorems on Algebraic Groups" (American Journal of Mathematics 78(2):404-443) [JSTOR link], and having trouble with some of the notation and language. In particular, since Rosenlicht admits arbitrary ground fields but is writing before Grothendieck's development of scheme theory, I'm confused by what he means by a point of an algebraic variety.
Rosenlicht is working with an algebraic group $G$ defined as "a union of a finite number of disjoint algebraic varieties [I presume "variety" means irreducible in this context] ... together with a group structure [defined by rational maps] ... a field $k$ is called a field of definition of $G$ if it is a field of definition for each component of $G$ and for all of the above rational maps..." (p. 402).
Question: In this context, what are the elements of $G$?
In particular if I want to think of $G$ as a separated scheme of finite type over $k$, is Rosenlicht talking about the closed points? Only the closed points with residue field $k$? Or the closed points of the base change to $\overline k$? Or what?