Consider four non-collinear points $A,A',B,B'$ in the projective space $\mathbb{R}P^2$. I want to show that the points $A,A',B,B'$ form a projective basis if and only if there exists a projective transformation $f$ with the properties: $f^2 = I_d$ and $f(A)= A'$, $f(B)= B'$.
I was thinking about looking for a specific form for $f$, but this became to difficult. So I took for the four points the standard projective basis $A = [(1,0,0)]$, $A' = [(0,1,0)]$, $B = [(0,0,1)]$ and $B' = [(1,1,1)]$. In this case, the $f$ with the properties would have the matrix form: \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 0 & 0 & -1 \end{pmatrix}
So we proved it here in the case of the standard projective basis and because you can always find a projective transformation between the standard basis and another basis, the statement must hold for every projective basis. But I think there should be a rigorous way of proving this.