inv_e_metric is the precision of a zero-mean normal distribution. This code is meant to sample from the corresponding normal distribution, which is my understanding of the correct linear algebra.

Eigen computes A=LL^T, so A.llt().matrixL().solve(u); computes L^{-1}u and if u\sim N(0,I), then \mathbb{E}(L^{-1}u(L^{-1}u)^T) = L^{-1}L^{-T} = (L^T L)^{-1}\neq A^{-1}.

@betanalpha I can do a bug fix this evening if you need.