I learned the way you described: Numbers -> Complex Numbers -> Vectors -> Matrices -> Quaternions -> SO(2) -> SO(3) -> Lie Algebras -> GA (The last 3 might ne

I learned the way you described: Numbers -> Complex Numbers -> Vectors -> Matrices -> Quaternions -> SO(2) -> SO(3) -> Lie Algebras -> GA (The last 3 might need air quotes around "learned") But I actually think it could be taught: Numbers -> Vectors -> GA -> Maybe Matrices? Discover complex number and quaternions naturally, by playing with rotations. Discover e the same way! (They taught me e through finance.) This would have explained so many of the weird quirks of math I had questions about. I bet it would have made relativity easier. Indeed, this isn't going to happen anytime soon.