" tommy quaternion "

[MphLee]

Hi, why you don't specify that \( C*B=A \) 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 \( \mathbb T \) is a \( k \)-vector space of dimension d=4 on which is defined a bilinear application \( *:{\mathbb T} \times {\mathbb T}\to {\mathbb T} \) that is not associative... but it must be bilinear because we want it distributive:

1. An element is of the form \( T=x1+yA+zB+wC \). Let  \( T,R,S,U\in{\mathbb T} \), the application being distributive means that
   
   \( (T+R)*(S+U)=T*S+R*S+T*U+R*U \)
   
   i.e. every translation by a Tommy quaternion is a endomorphism of  the addition group;
   
   
   
2. and we also want that for every scalar \( \lambda \)
   
   \( \lambda R*S= R*\lambda S=\lambda (R*S) \)
   
   i.e we want it to contain a copy of the base field, i.e multiplication by scalar  is multiplication by \( \lambda1 \) where \( 1\in \mathbb T \). So the previous means that the Tommy quaternions of the form \( T=x1+0A+0B+0C \) 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 \( T \) has a 4x4 matrix \( M_T \).
Obviously if it is not associative we "should not" have \( M_{T*S}=M_{T}M_{S} \) in general...

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