Which sort of explicit expression do you want? Michael Berlitzky had it fairly explicitly in his answer. Yokota has a description of the Z/2 action on SU(3) before Prop. 2.12.1, but I think he has a typo: it should be \epsilon(A) = (A^T)^(-1) (transpose inverse, combining two anti-automorphisms)

On further investigation, Theorem 3 of the 1947 Jacobson paper appears to be wrong, it omits the non-identity component of automorphisms of $h_n(\mathbb{C})$ (coming from an anti-automorphism of the matrix algebra rather than an automorphism).

As far as I can tell, Yokota is almost exclusively interested in the cases that give exceptional groups, right? Does that paper consider the other cases in your original question?

Thanks, I will look at that (and the other references you give)! Unfortunately for my application I need the other component for $h_3(\mathbb{C})$ too, which seems to have gone little-noticed in the literature.

I also need this result, and have been fairly frustrated looking for references: almost all references talk about the Lie algebra and not the Lie group of symmetries. An answer can be extracted from Michael Orlitzky's example, but it's hard for me to believe that this was not written down before April 2023.

So to specialize to $n=3$ translate into the notation for Lie groups that the question used, your answer is that $G_{\mathbb{R}} = PO(3) = SO(3)$, $G_{\mathbb{C}} = PU(3) \rtimes \mathbb{Z}/2$, and $G_{\mathbb{H}} = PSp(3)$?

I wanted a more precise reference, so I looked at Kac's book, and the fact that the dual Coxeter number appears as the ratio of Killing form to what you call the minimal form appears as Exercise 6.2 in Sec. 6.8. (I'd love another reference that talks about this more directly.)