How to swap multiple axis of a transformation matrix
Perhaps this is what you're looking for. We have $$ \begin{bmatrix}k_{z} & i_{z} & -j_{z} & -t_{z}\\ k_{x} & i_{x} & -j_{x} & -t_{x}\\ k_{y} & i_{y} & -j_{y} & -t_{y}\\ 0 & 0 & 0 & 1 \end{bmatrix} = \\ \begin{bmatrix} 0&0&-1&0\\ -1&0&0&0\\ 0&-1&0&0\\ 0&0&0&1 \end{bmatrix} \begin{bmatrix}i_{x} & j_{x} & k_{x} & t_{x}\\ i_{y} & j_{y} & k_{y} & t_{y}\\ i_{z} & j_{z} & k_{z} & t_{z}\\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 0&0&-1&0\\ -1&0&0&0\\ 0&1&0&0\\ 0&0&0&1 \end{bmatrix}. $$