Why rotations with two angles in $3D$ do not form a group? Let us use any parametrization of  $3D$ rotations with three angles (e.g. Euler angles or yaw-pitch-roll), and throw away one of the angles (just assign it a fixed value).
Will the remaining set of transformation form a group? If yes — which? If no — why?
Follow up: the same question on more general Lie groups. What typically happens if we fix some of the parameters? In which cases does this result in getting a new group? If it doesn't — why?
CLARIFICATION
Just in case ⁠— I'm NOT asking why the new set of transformations is not $SO(3)$ anymore, that's pretty obvious. The question is: which group axioms are no longer satisfied? We clearly have a neutral element, and for each transformation there is an inverse. So what's wrong then?

Let consider some 90 degree rotations.
We have two rotation matrices that generate our group.
$P=\pmatrix{&1&\\-1&&\\&&1}$ and $R=\pmatrix{1&&\\&&1\\&-1&}$
Traditionally our third rotation matrix
$Y=\pmatrix{&&1\\&1&\\-1&&}$ has been left out.
But $PRP^{-1} = Y$
Two rotations will generate the 3rd rotation and hence $SO_3$
