In the paper " THE COMPONENT GROUPS OF NILPOTENTS IN EXCEPTIONAL SIMPLE REAL LIE ALGEBRAS" by D. King I am unable to proof the lemma 3.7 which is omitted there. The lemma is following: Let $\mathfrak g $ be a exceptional and $R$ be a regular subgroup ( normalized by a maximal torus) of $G$. Let ($x_1,e_1,f_1$) and ($x_2,e_2,f_2$) be semi-regular triples in Lie($R$).(Semi-regular means centralizer of $(x_1,e_1,f_1)$ in $R$ is equal to the center of $R$) Now if ($x_1,e_1,f_1$) and ($x_2,e_2,f_2$) are conjugate in $G$ then they are conjugate in $R$.

It is also not clear to me that why he assume $\mathfrak g$ to be exceptional Lie algebra. Is it not true for classical Lie algebra.

Any reference for the proof of this lemma is also useful.