A four-vector is defined to be a four component quantity $A^\nu$ which transforms under a Lorentz transformation as $A^{\mu'} = L_\nu^{\mu'} A^\nu$, where $L_\nu^{\mu'}$ is the Lorentz transformation matrix, which including boosts, rotations and compositions. (In other words, as the components of a position vector $(x0,x1,x2,x3)$would transform).
The useful property of four-vectors is claimed to be that if two four-vector expressions are equal in one frame, then they will be equal in all frames :
$A^\mu = B^\mu \Leftrightarrow A^{\mu'} = B^{\mu'}$
and therefore we can express laws of physics in terms of four vectors, because they remain invariant in all frames.
But this property will be true even for four component quantities that transform (across reference frames) as $A^{\mu'} = T_\nu^{\mu'} A^\nu$, where $T$ is any transformation matrix (not necessarily a Lorentz one). As long as we can find a $T$ that will describe how the quantity's components transform, we can apply that T to both sides of an equality.
So why require (i.e. define) four-vectors to only be quantities that transform under a Lorentz matrix?
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