Given a projection matrix $P = [M | p_4]$, ($M \in 3 \times 3$, $p_4 \in 3 \times 1 $), the principal axis (the vector that passes through the center of projection and is perpendicular to the image plane) $v$ is $$v = det(M)m_3$$ where $m_3$ is the third row of $M$. Is this correct?
Now I want to find two more vectors, say $v_1$ and $v_2$, that form an orthonormal basis with $v$ and span the image plane. I know that $v_1$ and $v_2$ are not unique so I want them to be equal to the $x$- and $y$-axes once the principal axis coincides with the $z$-axis (of some world coordinate system). If the principal axis differs from the $z$-axis due to some transformation I want that same transformation to give me $v_1$ and $v_2$ if applied to the $x$- and $y$-axes. Hope you get what I mean by that!
Thanks in advance!