Hi, why you don't specify that in your list of identities? You forgot to add it or it is possible to derive it from the others?

Also, you see it as a non-associative algebra over the reals or the complex?

Trivial question (sorry but I'm new to linear algebra): being an algebra means that is a -vector space of dimension d=4 on which is defined a bilinear application that is not associative... but it must be bilinear because we want it distributive:

But this means that the multiplication is bilinear: every element should it should have a representation.. e.g. multiplication by has a 4x4 matrix .

Obviously if it is not associative we "should not" have in general...

Probably my understanding of non-associative alg. is so poor that I'm missing something obvious.

Also, you see it as a non-associative algebra over the reals or the complex?

Trivial question (sorry but I'm new to linear algebra): being an algebra means that is a -vector space of dimension d=4 on which is defined a bilinear application that is not associative... but it must be bilinear because we want it distributive:

- An element is of the form . Let , the application being distributive means that

i.e. every translation by a Tommy quaternion is a endomorphism of the addition group;

- and we also want that for every scalar

i.e we want it to contain a copy of the base field, i.e multiplication by scalar is multiplication by where . So the previous means that the Tommy quaternions of the form commute and "associate" with everything.

But this means that the multiplication is bilinear: every element should it should have a representation.. e.g. multiplication by has a 4x4 matrix .

Obviously if it is not associative we "should not" have in general...

Probably my understanding of non-associative alg. is so poor that I'm missing something obvious.