Joint Casimir Invariants? I'm a physicist so forgive me any improper use of terminology. This question is mostly me trying to find the correct math literature/terminology to learn about what I'm describing.
Suppose I have a Lie algebra $\mathfrak{g}$. This Lie algebra has some set of Casimir invariants, i.e. quantities that are conserved under the adjoint action by the corresponding Lie group.
Now consider the new Lie algebra $\mathfrak{g}\oplus\mathfrak{g}$. I am interested in identifying joint invariants under the action of the same Lie group element on each "block" of the direct sum. I.e. what is invariant under $(U^\dagger\oplus U^\dagger)(g_1\oplus g_2)(U\oplus U)$ where $g_1, g_2\in\mathfrak{g}$ and $U, U^\dagger$ are in the Lie group?
If it matters, I'm particularly interested in the Lie algebra $\mathfrak{su}(d)$ and the Lie group $SU(d)$. Mostly, I'm curious if there is mathematical literature on these sorts of invariants and if so, what terms I should be using to search for it.
 A: I've been rereading the examples you gave and I think I might be able to clear a few things up.
First things first, Casimir elements do not, in general, live in the Lie algebra but instead in its universal enveloping algebra $U(\mathfrak{g})$. This thing is a infinite dimensional algebra with an identity element with the same generating relations as the Lie algebra but no limits as it were. It has the neat property that every representation of $\mathfrak{g}$ extends to a representation of $U(\mathfrak{g})$.
Now the Casimir invariants live in the centre of $U(\mathfrak{g})$ but we can also interpret them as symmetric polynomials in the adjoint representation. The first and most obvious of these is the quadratic one. This is nothing but the  Killing form under a different guise. The "angle" between two elements of $\mathfrak{so}_n $ should be nothing more than the Killing form of those two elements and so that is preserved by the simple adjoint action (no diagonal action needed here). I believe in the Physics case this is related in some way to the total angular momentum although my physics knowledge here is sketchy at the very best.
