Does Extraction Support work with Lich's Eternal Life?

Hi,

Trying to understand the interaction for Extraction Support gem. The explanation states that the skills get 35% of damage as extra chaos, and as another line of text skills sacrifice 10% life on use.

When I insert the support gem, in-game dmg tooltip increases (both hit and dot). I have read that "extra dmg as" mods only works for hits though, so I am confused. Even for hit part, do I really get the extra dmg if Eternal Life prevents life to change?

Gem description does not mention any condition, like Minion Pact for example:
Extraction:
Supports Spell Skills which cause Damaging Hits. Using supported Spells will cause you to Sacrifice a portion of your Life on use, gaining extra Chaos Damage in exchange.
Supported Skills Gain 35% of Damage as extra Chaos Damage
Supported Skills Sacrifice 10% of Life on use

Minion Pact:
Supports damaging skills you use yourself, draining the life from one of your Minions on use in order to boost the skill's damage. Only takes effect if you have a Minion with enough life to drain the full amount.
Supported Skills remove 100% of your maximum Life from a nearby Minion on use to deal 30% more damage

It would be great if someone can explain the mechanics for me.

Cheers,