Why is $\mathbb{H}\otimes\mathbb{H}\cong\text{End}_\mathbb{R}\mathbb{H}$? When I first learned of the quaternion algebra $\mathbb{H}$, the most concrete way to get a grip on the ring of its endomorphisms $\operatorname{End}_\mathbb{R}(\mathbb{H})$ was to view them as $4\times 4$ matrices with real entries.
It was mentioned to me that $\operatorname{End}_\mathbb{R}(\mathbb{H})\cong\mathbb{H}\otimes_\mathbb{R}\mathbb{H}$ too, but this isomorphism is not as clear to me as the isomorphism to matrices. Is there a clear justification for this isomorphism with the tensor product? 
Thank you,

The algebra isomorphism sends $q_1 \otimes q_2$ to the endomorphism taking q to $q_1q\bar q_2$.  Not sure of the best way of showing it is injective (or equivalenty surjective by dimensionality).  One way is explicity computing the kernel using the tensor product basis of the standard 1,i,j,k basis of H.
