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.

An involution on a finite dimensional algebra $A$ over $\mathbb{Q}$ is an involutive anti-automorphism of $A$. If $\sigma$ is an involution on $A$, we say that $\sigma$ is positive if $\mathrm{Tr}_{A/\mathbb{Q}}(x\sigma(x))>0$ for all $0\neq x\in A$.

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 is well-known, but I do not know a reference for it. I have only found references to partial classifications.

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.