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Results ranging from Ashtekar variables to the perturbativeBern-Carrasco-Johansson (BCJ) double copy suggest a deep relation betweenYang-Mills theory and Einstein gravity. I examine this relation by writing downthe tetradic Palatini action for Einstein gravity and covariantly decomposingits 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.

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