in carbonic acid there is a hydrogen bonded moleculethen why it is not an organic compound

Carbonic acid is the name of compound having chemical formula H2CO3.This acid is formed when carbon dioxide dissolves in water.CO2 is also not an organic compound.Organic compounds are those compounds which have carbon-carbon and carbon-hydrogen bonds.But in carbonic acid there is neither carbon-carbon bond and nor carbon bonded to hydrogen.In fact in this compound carbon is bonded to oxygen atoms.The structure suggest that it is an in inorganic compound and not the organic one.The hydrogen atoms present in the compound are bonded to oxygen atoms and not with the carbon atom.