It seems that shadow imbue has a lucky hit chance. Does that mean that the lucky hit chance from flurry gets overriden with the shadow imbue LHC? Or does it check from both the flurry atk and the shadow imbue on it?
Shadow Imbuement has a Lucky Hit chance because it does its own instance of damage. If you attack with a Shadow Imbued skill like Flurry, you get one check when Flurry hits and then a separate check when Shadow Imbuement explodes. Note that Poison also has a Lucky Hit chance but Cold does not since it doesn’t deal any damage to trigger the Lucky Hit.
so with Flurry having 20% LHC. All 4 hits will check 4 times with a 20% LHC and on top of that, all 4 hits will again check 4 times with 33% LHC?
Not exactly. Skills that deal direct damage over multiple hits, like Flurry, list the total LHC for the skill across all hits. So a 20% LHC on the tooltip means that there is a 20% overall chance that at least one of the hits will proc a lucky hit. The chance per hit is much lower, closer to 5.4%.
Furthermore, Shadow Imbuement doesn’t deal its explosion damage right away*. Each hit of Flurry attempts to apply the Shadow Imbuement debuff. If all 4 swings hit the same enemy, the first one applies the debuff and the other 3 don’t apply anything since it’s already affected. If the enemy survives long enough, the debuff will trigger a LHC check when it deals its single-target damage. If the enemy dies while affected, the debuff’s explosion will instead damage nearby enemies and perform a LHC check on them.
*This gets weird when you use legendaries and uniques that trigger a Shadow Imbuement explosion and/or turn it into a DoT. I’m not sure about the math for all that, sorry 