A theorem of Albert [1] classifies finite dimensional division algebras over $\mathbb{Q}$ that admit a positive involution. I know how to deduce the classification of all finite dimensional simple algebras over $\mathbb{Q}$ that admit a positive involution from Albert's result, and this must be well-known, but I do not know a reference for it.

Is there a reference for this classification?

One reason this is not in the references I looked at is that when the motivation is the study of Abelian varieties, one can immediately reduce to the case of a division algebra.

[1] Albert, A. A. On the construction of Riemann matrices. II. Ann. of Math. (2) 36 (1935), no. 2, 376–394.