What's the difference between 3 and 4 quadrant triacs?
NXP has a good explanation as to what the differences between 3-quadrant and 4-quadrant triacs are.

Summary: a 3-quadrant triac cannot be turned on in the fourth quadrant (T2 negative, G positive) which is beneficial when dealing with inductive loads - ringing caused by sudden turn-off could potentially turn on a 4-quadrant triac, necessitating snubbers and other protection means to prevent spurious conduction.