Multiplication of vectors. That’s the meaning of the algebra word in the phrase, finite dimensional associative division algebra. In R^3 you can’t define mult

Multiplication of vectors. That’s the meaning of the algebra word in the phrase, finite dimensional associative division algebra. In R^3 you can’t define multiplication of vectors in a way that works. That is, in a way that each vector can be multiplied by itself and in which each vector has a multiplicative inverse while preserving the real vector space structure.