  We compute the fundamental class (in the extended Bloch group) for
representations of fundamental groups of 3-manifolds to SL(4,R) that factor
over SL(2,C), in particular for those factoring over the isomorphism PSL(2,C) =
S0(3,1). We also discuss consequences for the number of connected components of
SL(4,R)-character varieties, and we show that there are knots with arbitrarily
many components of vanishing Chern-Simons invariant in their SL(n,C)-character
varieties
