For any metric $$g_{μν}$$ is there always a linearly independant spacetime algebra satisfying $$\{\bar{γ}_μ,\bar{γ}_ν\} = 2 g_{μν} I?$$
For a diagonal metric I was able to work out that $$\bar{γ}_μ=\sqrt{n_{μμ}*g_{μμ}}γ_μ$$ satisfied these conditions (the minkowski simply adds negatives to cancel the spacelike gammas). However for metrics which are not diagonalizable at every point in spacetime I've been having trouble.
$$+---$$ is being used here.
Playing around with Tetrads seems like the way to go but I havent had as much luck this far. Thanks in advance to any help!