I'm looking for an expression for the oblique projection of the row space of a matrix $A$ along the row space of the matrix $B$ onto the orthogonal complement of $C$. Simply modifying the well-known oblique projection formula for the oblique projection of A along $B$ onto $C$
$A/_BC=[A/B^\perp][C/B^\perp]^\dagger C$ (or other/similar expressions)
turns out to be not that easy...
The notation $A/B$ means the orthogonal projection of $A$ onto $B$ and can be written as
$A/B = AB^\top(B B^\top)^\dagger B = A\Pi_B$.
with $\Pi_B$ as the projection matrix. $A/B^\perp$ means the orthogonal projection onto the orthogonal complement of $B$ and can be expressed as
$A/B^\perp = A(I-\Pi_B)$
with $I$ as an appropriate identity matrix. The superscript $\dagger$ means the Moore-Penrose pseudo inverse. The upper expression $A/_BC$ can also be written as
$$A/_BC=A \left[ \begin{array}{cc} C&B \end{array} \right] \left[\left[ \begin{array}{cc} CC^\top&CB^\top\\ BC^\top&BB^\top \end{array} \right]^{-1}\right]_{first~r~columns} C $$
with r the row dimension of $C$. What I'm looking for is a expression for $A/_BC^\top$.
Does someone have at least an idea how I could find such an expression?