Remark 1

Tag 00M2, permanent. Cite it as [Bourbaki, Tag 00M2].

Algebra, VIII, p. 368, § 19. Quaternion Algebras

Let O be the octonion algebra of type $(-1,0,-1,-1)$ over R (III, Appendix, No. 3, p. 615). Let D be an alternative Cayley algebra over R such that every nonzero element of D has an inverse. We can prove (VIII, p. 370, Exercise 5) that D is isomorphic to R, C, H, or O.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Exercise 5, § 19

Cites

III, Appendix, No. 3, p. 615
VIII, p. 370, Exercise 5