How to prove the tensor product over $\mathbb{R}$ of two copies of the quaternions is isomorphic to the matrix algebra $M_4 (\mathbb{R})$ as algebras over $\mathbb{R}$? More precisely, the problem is to show the isomorphism $\mathbb{H} \otimes_\mathbb{R} \mathbb{H} \cong M_4 (\mathbb{R})$.
On the book "Spin Geometry" by Lawson and Michelsohn, page 27, there is an isomorphism defined by sending $q_1 \otimes q_2$ to the real endomorphism of $\mathbb{H}$ which is given by $x \mapsto q_1 x \bar{q_2}$, but I don't know how to deduce that this real algebra homomorphism is in fact an isomorphism.
Best Answer
It's a basic fact (here's a proof in the second proposition on page 157) that the tensor product of two central simple algebras is another central simple algebra. A proof should be available wherever central simple algebras are discussed.
Another location in Jacobson's Basic Algebra II on page 218-219.
Another location in Rowen's Ring theory Theorem 1.7.27.
Another location in notes by Morandi.
By simplicity of the ring, the kernel of your (nonzero) ring homomorphism is automatically $\{0\}$, showing it is injective.
Finally, since the image and codomain are both 16-$\Bbb R$-dimensional, they are equal, showing the must also be surjective.