Real algebraic subgroups action in the projective space

Question

Let $G$ be an abelian closed normal real algebraic subgroup of the complex projective linear group. Let $H$ be a closed subgroup of $G$. If the homogeneous space $X:=G/H$ is contained in the complex projective space.

1. Is the action of $G$ on $X$ linear (algebraic)?

2. Is $X$ closed in the complex projective space?