and $\operatorname{\mathrm{Aut}}(\Delta)$ be the automorphism group of $\Delta$. A subgroup $G$ of $\operatorname{\mathrm{Aut}}(\Delta)$ is said to be **strongly transitive** if $G$ is transitive on the set of all pairs $(A,C)$ where $A$ is an apartment of $\Delta$ and $C$ is a chamber of $A$. If $\Delta$ is a usual (non-topological) building, we say that $\Delta$ is **strongly transitive** when $\operatorname{\mathrm{Aut}}(\Delta)$ is strongly transitive. If $\Delta$ is a compact building, this terminology is rather used when $\operatorname{\mathrm{Auttop}}(\Delta)$ is strongly transitive.

The following fact is almost immediate but worth mentioning.

\[lemma:strongtransitivity\] Let $\Delta$ be a spherical building and $\pi, \pi'$ be opposite panels of $\Delta$. Let also $G$ be a strongly transitive subgroup of $\operatorname{\mathrm{Aut}}(\Delta)$. Then the stabilizer $\operatorname{\mathrm{Stab}}_G(\pi, \pi')$ of $\pi$ and $\pi'$ in $G$ acts $2$-transitively on $\operatorname{\mathrm{Cham}}(\pi)$.

Let $C_1 \neq D_1$ and $C_2 \neq D_2$ be chambers in $\operatorname{\mathrm{Cham}}(\pi)$. We want to find $g \in \operatorname{\mathrm{Stab}}_G(\pi, \pi')$ such that $g(C_1) = C_2$ and $g(D_1) = D_2$. Let $A_1$ (resp. $A_2$) be the apartment of $\Delta$ containing $C_1$ (resp. $C_2$), $D_1$ (resp. $D_2$) and $\pi'$. By strong transitivity of $G$, there is $g \in G$ such that $g(A_1) = A_2$ and $g(C_1) = C_2$. Clearly, we also have $g(\pi) = 