We have a 4-vector $V^\mu =(V,0,0,V)$ with sametime and $ z $ components. then we are asked to find the transformations that are not pure rotations and leave $ V $ invariant. Sone one such matrix is the rotation matrix as follows:$$ \Lambda^\mu _\nu=\begin{pmatrix}1&0&0&0\\0&cos\theta &sin\theta&0\\0&-sin\theta &cos \theta &0\\0&0&0&1 \end{pmatrix}$$ The other matrix is $$K_3 =\begin{pmatrix}0&0&0&1\\0&0 &0&0\\0&0 &0 &0\\1&0&0&0 \end{pmatrix}$$ and $K_3 +J_3$ where $J_3$ is just the rotation matrix in $xy$ plane with an angle of 90 degree.

How do I go about finding the three generators of my two transformations and a rotation? Thank you.


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