(a+b)+c=a+(b+c)	associative OK
a+b=b+a	commutative addition OK
a⋅(b+c)=a⋅b+a⋅c	distributive ????
a+0=a	identity OK
a+(-a)=0	inverse OK
NOT a⋅b=b⋅a (well, the dot product over vectors over the body of real numbers is.)
NOT a⋅(b⋅c)=(a⋅b)⋅c	OOPS
NOT a⨯b=b⨯a