The O(3,1) Yang-Mills equations and the Einstein equations

The vacuum Einstein equations (with cosmological constant) written in a slightly unconventional manner, can be decomposed into three parts: the first two parts are the ordinary self dual Yang-Mills equations and the anti-self dual Yang-Mills equations for anO(3,1) gauge group, on an unspecified background space-time, the third part are equations that solder or relate these two Y-M fields and connections to the curvature and connection of that unknown space-time. It is the purpose of this note to take this point of view seriously and concentrate on the first two parts in their own right. We apply to them generalizations of solution construction techniques which have arisen from the study of self dual Yang-Mills equations on Minkowski space. At the end we discuss how to solder or bootstrap these results to the determination of the space-time itself.