Theorem 4

(i) Let $P_1$ and $P_2$ be two parabolic subgroups of $G$ whose intersection is parabolic and let $g \in G$ be such that $gP_1g^{-1} \subset P_2$. Then $g \in P_2$ and $P_1 \subset P_2$.

(ii) Two parabolic subgroups whose intersection is parabolic are not conjugate.

(iii) Let $Q_1$ and $Q_2$ be two parabolic subgroups of $G$ contained in a subgroup $Q$ of $G$. Then any $g \in G$ such that $gQ_1g^{-1} = Q_2$ belongs to $Q$.

(iv) Every parabolic subgroup is its own normaliser$^6$.

Assertion (i) follows from Props. 3 and 4, and implies (ii). Under the hypotheses of (iii), we have $gQ_1g^{-1} \subset Q$, which implies that $g \in Q$ by (i). Finally, (iv) follows from (iii) by taking $Q_1 = Q_2 = Q$.