A complex Lie group may have several real forms.
Are there any duality/trinity... between them?
Maybe a trivial question to ask, is $SL(3,\mathbb{C})$ a real form of $SL(3,\mathbb{C})\times SL(3,\mathbb{C})$ ?



Any duality between different real forms of a complex Lie group?
I think not. For a compact connected real Lie group $G$, the real forms of the complexification of $G$ are classified by the set of elements of order 2 in a maximal torus of $G$ modulo the action of the Weyl group. This set appears to have no obvious symmetries. (The quoted result can be found, for example, in Serre's Galois Cohomology).


