Remark

If $k=\mathbf{C}$, the subalgebras $\mathfrak{g}^1(a)$, for $a$ regular and belonging to a connected component M of G, are conjugate under Int($\mathfrak{g}$). Indeed, let R be the set of regular elements of G. For all $a\in R\cap M$, let $M_a$ be the set of those $b\in R\cap M$ such that $\mathfrak{g}^1(a)$ is conjugate to $\mathfrak{g}^1(a)$ under Int($\mathfrak{g}$). We have Int($\mathfrak{g}$) $=$ Ad(G$^0$), where $G^0$ is the identity component of G. By the Corollary to Prop. $5, M_a$ is open in R. It follows that $M_a$ is open and closed in R. Since $k=\mathbf{C}, R\cap M$ is connected (Lemma 1), hence $M_a= R\cap M$.