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.

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