I am an undergraduate in physics and know little about math. I know about some basic ideas of Lie groups and Lie algebras like roots, weyl group, weyl chambers but I am ignorant about complexification, symmetric space and so on. When I was reading the paper A geometric theory of non-local two-qubit operationshttps://arxiv.org/abs/quant-ph/0209120, I was confused with the maths the author used.
It says at the bottom of page8 in this paper "From the Lie group representation theory, the orbits of local gates $K$ acting on $SU(4)/SU(2)\otimes SU(2)$ are in one-to-one correspondence with the orbits of the Weyl group $W(G,K)$ on $\mathfrak{a}$". (the ref is differential geometry, Lie groups and symmetric spaces) But I didn't find the theorem in this book. It may be in Chap7. Actually, I didn't know the definitions of those two actions.
What are the definitions of those two actions? Where can I find this theorem?