Weyl group of semisimple symmetric space

This paper investigates a generalization of the notion of the Weyl group of a real reductive Lie group to a semisimple symmetric space $G\mathrm{/}H$. First, we show that the group structure of the Weyl group of $G\mathrm{/}H$ is independent of the choice of split Cartan subalgebras of $G\mathrm{/}H$ and the choice of Cartan involutions of $G$ stabilizing $H$. After that, we can understand all elements of the Weyl group of $G\mathrm{/}H$ by comparing the Weyl group of some reductive Lie group associated to $G\mathrm{/}H$. With the aim of extending these results to reductive real spherical homogeneous spaces, this work serves as a first step in this direction.