  We revisit the problem of performing conformal block decomposition of
exchange Witten diagrams in the crossed channel. Using properties of conformal
blocks and Witten diagrams, we discover infinitely many linear relations among
the crossed channel decomposition coefficients. These relations allow us to
formulate a recursive algorithm that solves the decomposition coefficients in
terms of certain seed coefficients. In one dimensional CFTs, the seed
coefficient is the decomposition coefficient of the double-trace operator with
the lowest conformal dimension. In higher dimensions, the seed coefficients are
the coefficients of the double-trace operators with the minimal conformal
twist. We also discuss the conformal block decomposition of a generic contact
Witten diagram with any number of derivatives. As a byproduct of our analysis,
we obtain a similar recursive algorithm for decomposing conformal partial waves
in the crossed channel.
