Example of a non-associative algebra
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The multiplication is defined by taking the complex conjugate of the usual multiplication: . This is a commutative, non-associative division algebra of dimension 2 over the reals, and has no unit element.
Proof that is a division algebra
It remains to prove that the binary operation given above satisfies the requirements of a division algebra
- (x + y)z = x z + y z;
- x(y + z) = x y + x z;
- (a x)y = a(x y); and
- x(b y) = b(x y);
for all scalars a and b in and all vectors x, y, and z (also in ).
(similarly for right distributivity); and for the third and fourth requirements
Non associativity of 
So, if a, b, and c are all non-zero, and if a and c do not differ by a real multiple, .