The set of Dirac spinors is a representation of the Lorentz algebra, whose generators are represented by the Dirac matrices $\gamma^\mu$ via the commutators, ie an element of the the Lorentz group is $\exp\left(\frac{1}{2}[\gamma_\mu, \gamma_\nu]\omega^{\mu\nu}\right)$ where $\omega^{\mu\nu}$ are the parameters.

The Dirac adjoint $\bar{\psi}=\psi^\dagger\gamma_0$ is defined such that the bilinear form $\bar{\psi}\psi$ is an invariant under transformations of the Lorentz group.

In the theory of compact Lie groups, we know there is a unique measure which induces a unique bilinear form on the Lie algebra.

Is this the same case for the Lorentz group? If not what are the other invariants? Is this the case for all (spin-) representations of the Lorentz group?