How do I cut large lilac branches?

Best to cut 6" away from where the branch meets the trunk to remove the weight, then make a second cut close to the truck, that way you don't damage the trunk.

Wait until the bush goes dormant in the fall and you can prune away the branches . I'm not a tree expert but I have always pruned my trees & bushes during the fall and they come back every spring beautifully. I'm sure you can find how to videos on the web if you need further instructions. Good luck

I've had great luck cutting big branches after lilacs are done flowering. Shrubs do well with a good pruning. Just make sure you cut less than a third of the total number of them so it stays healthy.

All of the above. Major pruning should be done in a dormant time. But ducking under to walk should be taken care of now. As Sharon said, prune back the offending limb until you've just got a relatively lightweight piece of branch. Start cutting from below so the bark won't peel or rip and then go ahead and cut from the top. Use a good, clean tree saw blade. You can cut as close as a couple of inches from the main trunk. But, since this is a big piece, I would also recommend sealing it with tree sealer to prevent disease and insect invasion. Love those lilacs!