Let $G=SL_3(\mathbb{C})$ and $X=G/B$ be the associated full flag variety. Fix a non-degenerate symmetric quadratic form $Q$ on $\mathbb{C}^3$. This gives an order $2$ automorphism $F_Q$ of $X$, mapping a flag $(L,P)$ to $(P^{\perp},L^{\perp})$. Is there any explicit description of the quotient $X/F_Q$ ?

(Remark that if $Q$ is non-degenerate, $F_Q$ is fixed point free.)

Let $G=SL_3(\mathbb{C})$ and $X=G/B$ be the associated full flag variety. Fix a non-degenerate symmetric quadratic form $Q$ on $\mathbb{C}^3$. This gives an order $2$ automorphism $F_Q$ of $X$, mapping a flag $(L,P)$ to $(P^{\perp},L^{\perp})$. Is there any explicit description of the quotient $X/F_Q$ ?

Let $G=SL_3(\mathbb{C})$ and $X=G/B$ be the associated full flag variety. Fix a non-degenerate symmetric quadratic form $Q$ on $\mathbb{C}^3$. This gives an order $2$ automorphism $F_Q$ of $X$, mapping a flag $(L,P)$ to $(P^{\perp},L^{\perp})$. Is there any explicit description of the quotient $X/F_Q$ ?

Let $G=SL_3(\mathbb{C})$ and $X=G/B$ be the associated full flag variety. Fix a non-degenerate symmetric quadratic form $Q$ on $\mathbb{C}^3$. This gives an order $2$ automorphism $F_Q$ of $X$, mapping a flag $(L,P)$ to $(P^{\perp},L^{\perp})$. Is there any explicit description of the quotient $X/F_Q$ ?

(Remark that if $Q$ is non-degenerate, $F_Q$ is fixed point free.)