In his book *Linear Algebraic Groups*, Tonny Springer defines a **Borel subgroup** of a linear algebraic group $G$ over an algebraically closed field to be a "closed, connected, solvable, subgroup of $G$, which is maximal for these properties".

Apart from having too many commas, this definition always bugged me for the following reason: if we take the closure of a connected solvable subgroup $H \subseteq G$, don't we get a connected solvable subgroup $\overline{H}$ with $H \subseteq \overline{H}$? If so, we wouldn't need to include the word "closed" in the definition of Borel subgroup: a maximal connected solvable subgroup would automatically be closed.

Now that I see the same definition on GroupProps, I feel I must be making a mistake. If so, what is it?

And while I'm at it: is there a generally accepted definition of 'Borel subgroup' for affine group schemes over a field that's not necessarily algebraically closed?