Assume a four-vector in the most general case in frame $\Sigma$: $$ \begin{bmatrix} t \\ x(t,y,z) \\ y(t,x,z) \\ z(t,x,y) \end{bmatrix} $$ This can be a four-velocity, four-acceleration or other things. (given that the derivatives of $x,y,z$ with respect to time or anything else would result in a function of those again this can be interpreted as any general four-vector). Now if the frame $\Sigma'$ is moving with velocity $\vec v$ in three dimensional space with respect to the initial frame, what is the lorentz transformation resulting $t',x',y',z'$?
NOTE: I don't want to rotate the initial system so that its x-axis aligns with the direction of the second frames velocity. I would like to have it a general way.
Can such transformation exist? And to be more general what are the steps of finding a lorentz transformation for a given four-vector?
I came across a general boost function but that seems to be only for four-velocity.