What is a Borel subgroup of a linear algebraic group, or affine group scheme? 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?
 A: It's an algebraic subgroup of $G$, not an abstract subgroup of the points $G(k)$ of $G$. In the old days, when algebraic groups were smooth over algebraically closed fields and $G$ was identified with its points $G(k)$, authors used "algebraic subgroup" and "closed subgroup" interchangeably. What Springer really means is that a Borel subgroup is a (smooth) connected solvable algebraic subgroup of $G$ that is maximal for these properties.
For a smooth algebraic group scheme over an arbitrary field $k$, a Borel subgroup is usually defined to be an algebraic subgroup that becomes Borel over the algebraic closure of $k$. Alternatively, one can say that it is a smooth connected solvable algebraic subgroup $B$ such that $G/B$ is complete. This definition works well with respect to change of base field, and has the advantage of avoiding mentioning algebraic closures. 
To put it another way, the "closed" in Springer's definition isn't redundant because he was thinking only of closed subgroups. 
[I agree with YCor that the footnote was obviously written by Borel, and I find it insulting to Borel to suggest otherwise.]
