TITLE: Tensor product between quaternions and complex numbers.
QUESTION [6 upvotes]: Let $H$ be the Hamiltonian quaternions, $\mathbb C$ the complex numbers and $\mathbb R$ the real numbers. Identify $H\otimes_{\mathbb R} 
 \mathbb C$ in familiar terms.
This is an exercise of Modern Algebra of Garrett Birkhoff. I think it may have something to do with the Octonions. Any ideas?
any ideas?

REPLY [4 votes]: The quaternions $\mathbb{H}$ are a right vector space over $\mathbb{C}$. 
Then there is an algebra homomorphism $\phi:\mathbb{H}\otimes\mathbb{C}\to\mathrm{End}_{\mathbb{C}}(\mathbb{H})$ via
$$\phi(p\otimes z)q=pqz. $$
Using $\{1,\mathbf{j}\}$ as a basis for $\mathbb{H}$ over $\mathbb{C}$ this allows us to identify basis elements of $\mathbb{H}\otimes\mathbb{C}$ with matrices inside $M_2(\mathbb{C})$ in order to check injectivity of $\phi$ hence that it's an isomorphism by dimensions.