No, 5 atoms. See below.
The problem with having connected constraints is that the system of linear equations become quickly overdefined and then has no unique solution. This already happens if you have, e.g. two bonds and an 180 degree angle.
The options you have is 1 central atom with 1, 2, 3, or 4 bonds connected to it that are constrained, or two bonds and an angle where the middle atom of the angle is the central atom of the cluster (internally, there is no angle constraint, but the system becomes a triangle of 3 bonds)
None of the bond atoms may be the central atom for another SHAKE cluster.
So, you could have a shake constraint for all for S-O bonds, but not on the O-O distances or O-S-O angles.
To keep such a construct rigid in LAMMPS, you must use fix rigid or fix rigid/small, where you would assign a unique molecule ID to each \textsf{SO}_4^{2-} anion and then use the “molecule” keyword to define the rigid clusters.