We consider a natural almost complex distribution on the associated
bundle $F^{\left(n\right)}M$ to the principal bundle of the g-orthogonal
oriented frames on a Riemannian manifold (M, g), with standard fibre
$\frac{SO\left(2n+k\right)}{U\left(n\right)\times SO\left(k\right)}$:
we find necessary and sufficient conditions ensuring that the distribution
is an almost complex foliation in $F^{\left(n\right)}M$ and we compute
the Nijenhuis tensor. Finally, we characterize the local sections
of $F^{\left(n\right)}M$.
