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I am writing something on the complexification of a real associative algebra. There are two well-known isomorphisms about quaternions: $$\mathbb{H}\otimes_\mathbb{R}\mathbb{C}\simeq M_2(\mathbb{C})$$ and $$\mathbb{H}\otimes_\mathbb{R}\mathbb{H}\simeq M_4(\mathbb{R}).$$ I need a reference which includes both of these two isomorphisms. I also wonder if there is a reference dealing with the bimodules over the real division rings and their complexification. Anything about complexification and related Galois descent are also welcome! Thanks a lot!

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What you are looking for is contained in

T. Y. Lam: Introduction to quadratic forms over fields, Graduate Studies in Mathematics 67. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-1095-2/hbk). xxi, 550 p. (2005). ZBL1068.11023.

In particular:

  • the isomorphism $$\mathbb{H}\otimes_\mathbb{R}\mathbb{C}\simeq M_2(\mathbb{C})$$ is a consequence of the proof of parts (3), (4) of Proposition 1.1 (taking, in the Author's notation, $F=\mathbb{R}$, $E=\mathbb{C}$, $a=b=-1$), see the beginning of p. 53;
  • the isomorphism $$\mathbb{H}\otimes_\mathbb{R}\mathbb{H}\simeq M_4(\mathbb{R})$$ is the content of Exercise 9 p. 76, with $F=\mathbb{R}$.
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  • $\begingroup$ Thanks! This helps a lot. Do you know something which is more specialized on real algebras and complexification? $\endgroup$ – yzl Nov 2 at 3:43
  • $\begingroup$ You are welcome. At the moment, I have no other reference in mind... $\endgroup$ – Francesco Polizzi Nov 2 at 9:12

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