Perhaps you are
Embraced by rock, then taken by the air
Born unto water, yet a friend of fire
Dragons are often shown as living in caves (embraced by rock); they fly (taken by the air); many culture's dragons, like the biblical "Leviathan", are sea creatures (born unto water); dragons in Western mythology are usually fire-breathing (a friend of fire).
Child of the gods of the sky
Chinese dragons, in particular, are divine creatures, often particularly associated with gods of rain and thunder.
As mighty as the rock
As a fast as the wind
As wise as water
As impulsive as fire
Dragons are generally depicted as mighty, fast, wise, and (sometimes) impulsive or at any rate quick to anger.
Smoldering sky, and soggy soil
I kill with a keratin coil
Fire breathed down from above (smoldering sky); association with sea and caves (soggy soil); aside from their fiery breath, sharp teeth, and (in some cultures) magical powers, dragons are also typically shown as having sharp curved talons, which are presumably made of keratin. ("Coil" seems a bit of a stretch, though, so that's a weak point.)