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In a recent paper, we show that an S-matrix functional for gravitonamplitudes can be described by an N=8 supersymmetric gravitational holonomyoperator in twistor space. In this paper, we obtain an alternative expressionfor the gravitational holonomy operator such that it can be interpreted as asquare of an N=4 holonomy operator for frame fields, by taking a sum of certainshuffles over ordered indices. The new expression leads to amplitudes of notonly spin-2 gravitons but also spin-0 massless particles. We discuss that thesquared model is favored as a theory of quantum gravity.

Holonomies of gauge fields in twistor space 3: gravity as a square of N=4 theory