I am writing about the first homework question in Srednicki Chapter 83 (83.1 part a). Please help me check my understanding. In other words, would my reasoning below be right? Forgive me too if this is elementary and (if I'm wrong) please explain it for someone who has not had any group theory save for what he picked up in Srednicki along the way.
In going from the color group $SU(3)$ to $SO(3)$ for this chiral Lagrangian, the Weyl fields are now real (or a real representation?) and thus the flavor symmetry no longer has a L and R difference, i.e. it's no longer a (say there were two massless fields like before) $U(2)_L \times U(2)_R$ and now it's just $U(4)$.
Further, in order to avoid the anomalous axial $U(1)$ symmetry we need to make the group $SU(4)$ so that we restrict further than just unitary matrices: we go from matrices where the determinant has absolute value 1 to ones where the absolute value is a real 1.