Blindsight spot.

I just realised that in any unital ring commutativity of addition follows from distributivity:

	\(a + b\)
\(=\)	\(-a + a + a + b + b - b\)
\(=\)	\(-a + a\cdot(1 + 1) + b\cdot(1 + 1) - b\)
\(=\)	\(-a + (a + b)\cdot(1 + 1) - b\)
\(=\)	\(-a + (a + b)\cdot 1 + (a + b)\cdot 1 - b\)
\(=\)	\(-a + a + b + a + b - b\)
\(=\)	\(b + a\)

The same holds for unital modules, algebras, vector spaces, &c. Note that multiplication doesn't even need to be associative. It's amazing how such things can pass unnoticed.