The homotopy Lie algebra of classifying spaces

Let X be a 1-connected CW-complex of finite type and Lx its rational homotopy Lie algebra. In this work, we show that there is a spectral sequence whose E2 term is the Lie algebra ExtULx(Q, Lx), and which converges to the homotopy Lie algebra of the classifying space B autX. Moreover, some terms of this spectral sequence are related to derivations of Lx and to the Gottlieb group of X.