In theoretical physics, the electroweak model breaks parity maximally. All its fermions are chiral Weyl fermions. Some theorists found this objectionable, and so proposed a GUT extension of the electroweak group

to

.

Here, SU(2)_L (pronounced SU(2) left) is none other than SU(2)_W and B-L is the baryon number minus the lepton number.

There is also the chromodynamic SU(3)_C. The idea was to restore parity by introducing a left-right symmetry. This is a group extension of Z₂ (the left-right symmetry) by

to the semidirect product

Z₂.

This has two connected components where Z₂ acts as an automorphism, which is the composition of an involutive outer automorphism of SU(3)_C with the interchange of the left and right copies of SU(2) with the reversal of U(1)_B-L. This left-right symmetry is spontaneously broken to give a chiral low energy theory.

In this setting the chiral quarks

(3,2,1)_1/3

and

are unified into an irrep

.

The leptons are also unified into an irrep

.

This then predicts three sterile neutrinos , which is perfectly consistent with current neutrino oscillation data.

Because the left-right symmetry is spontaneously broken, left-right models predict domain walls.

Later, this left-right symmetry idea was used in the Pati-Salam model and trinification.