Einstein gravity as a theory with a SL(2,C) connection double copy

Results ranging from Ashtekar variables to the perturbative Bern-Carrasco-Johansson (BCJ) double copy suggest a deep relation between Yang-Mills theory and Einstein gravity. I examine this relation by writing down the tetradic Palatini action for Einstein gravity and covariantly decomposing its variables into two $SL(2,\mathbb{C})$ connections and two soldering forms. This leads to a conjecture that the BCJ double copy can be understood through a "holomorphic trail", a series of injection maps between two copies of $SU(2)$ theories and Einstein gravity.