Minimal parabolic k-subgroups acting on symmetric k-varieties corresponding to k-split groups

Symmetric k-varieties are a natural generalization of symmetric spaces to general fields k. We study the action of minimal parabolic k-subgroups on symmetric k-varieties and define a map that embeds these orbits within the orbits corresponding to algebraically closed fields. We develop a condition for the surjectivity of this map in the case of k-split groups that depends only on the dimension of a maximal k-split torus contained within the fixed point group of the involution defining the symmetric k-variety.