Why $y'=y$ and $z'=z$ in Lorentz transformation?

According to Resnick’s Introduction to Special Relativity,Lorentz transformation in general can be given as:
$x’=a_{11}x+a_{12}y + a_{13}z+a_{14}t$

$y’=a_{21}x+a_{22}y + a_{23}z+a_{24}t$

$z’=a_{31}x+a_{32}y + a_{33}z+a_{34}t$

$t’=a_{41}x+a_{42}y + a_{43}z+a_{44}t$

Firstly, I am unable to make a symmetry argument to prove that $y=y’$ and $z=z’$. I could prove $x’$ should be independent of $y$ and $z$.

Next, I disagree with the mathematical derivation of this in Resnick.

If $S$ and $S’$ are two frames with $S’$ moving with velocity $vec v$ relative to $S$, then without loss of generality we can orient $vec v$ to be along $X$ axis (for $S$) and $X’$ axis (for $S’$).

Then to prove $y=y’$ and $z=z’$ he uses an argument that $X’Y’$ plane and $XY$ plane should be the same. But just because their velocity vectors are along the X axis, their XY planes need not be the same right. . . the Y’ and Z’ axis could lie anywhere in the YZ plane as shown in fig.
So how can we prove $y=y’$ and $z=z’$?

Question:
What symmetry argument can be used to prove that $y=y’$

is it true that we can derive y=y’ and x=x’ without assuming any two axes to be aligned?

I know we can align one more axis (say z along z’) and prove the invariance… But isn’t it wrong otherwise?