I suppose this is a question directed at supergravity experts. Basically, I am a mathematician studying things related to *topological* string theory and in my work, the *Cremmer-Julia invariant* appeared. This is something in the physics literature in the world of $\mathcal{N}=8$ supergravity in four dimensions. My questions are basically due to my struggle in understanding the physics in these papers. I'm trying to understand if it appeared in my work coincidentally or accidentally, or if rather it is there for a deep reason.

A nice brief source is (https://arxiv.org/pdf/0804.0651.pdf) where the Cremmer-Julia (CJ) invariant appears in (3) and (22). From what I can gather, the CJ invariant is the unique quartic invariant under $E_{7(7)}$, which is the split form of $E_{7}$ Lie group. It is also described as being a defining invariant of U-duality in string theory. My questions are the following:

1. I know the split form $E_{7(7)}$ is the unique split form in the Lie group $E_{7}$, but I'm wondering if someone can help me understand more concretely what this is. Does it have a more explicit description?

2. Is there a nice mathematical description of exactly *what* is invariant here under $E_{7(7)}$ without reference to charges in a supergravity theory? I mean, is there some lattice or something which is preserved such that the CJ invariant is like related to a theta series of the lattice or something?

3. Finally, it would be great if someone could take a crack at explaining U-duality and its relation to $E_{7(7)}$ and the Cremmer-Julia invariant.