Are you saying my Thanatological/Tranference link with my zombie lets it "borrow" some of my life force?

So a lich is a lifeless creature animated by a (presumably) thanatological link with some other force, who in turn animates undead with thanatological links.

YO DAWG WE HERD U LIKE UNDEAD

I think it's technically "alive" because a critter can't be both a construct and undead under current mechanics.

There's a thanatological link keeping it alive through a biological medium (blood), but that doesn't necessarily mean anything. It certainly doesn't feel pain and doesn't even display basic animal intelligence.

And ignore all that stuff about them occasionally having a mind of their own and acting outside of the parameters of the spell. That's all just part of what we call the Ghost in the Spell Matrix.