  We find an explicit expression for the cross-covariance between stochastic
integral processes with respect to a $d$-dimensional fractional Brownian motion
(fBm) $B_t$ with Hurst parameter $H>1/2$, where the integrands are vector
fields applied to $B_t$. It provides, for example, a direct alternative proof
of Y. Hu and D. Nualart's result that the stochastic integral component in the
fractional Bessel process decomposition is not itself a fractional Brownian
motion.
