Noncommutativity of observables is a defining feature of quantum physics. It plays a fundamental role in the formulation of the uncertainty principle for complementary variables and strongly affects the thermodynamic behavior of systems with noncommuting, that is, non-Abelian, conserved quantities. We here derive extensions of the first and second laws of thermodynamics, including nonequilibrium fluctuation relations, for driven quantum systems. We identify an additional non- Abelian contribution to the energy and entropy balances, without which these relations would be violated. This contribution can be harnessed to enhance both work extraction and nonequilibrium currents compared to what is achieved in commuting thermodynamics. These results demonstrate that noncommutativity is a potentially valuable thermodynamic resource.