  We consider certain rational homotopical conditions of simly connected CW
complex $X$ such that the rational cohomology of the classifying space
$Baut_1X$ for fibrations with two-stage fibre $X$ is (not) free. First, we
consider when is $Baut_1X$ a rational factor of $Baut_1(X\times S^n)$ for an
odd-integer $n$ and observe for a non-formal elliptic space $X$ of rank 3.
Second, we compute the Sullivan minimal models of $Baut_1X$ when $X$ are
certain non-formal pure spaces of rank 5.
