  The $S=1/2$ spin ladder compound (C$_5$H$_{12}$N)$_2$CuBr$_4$ (BPCB) is
studied by means of high-resolution inelastic neutron scattering. In agreement
with previous studies we find a band of triplet excitations with a spin gap of
$\sim0.8$~meV and a bandwidth of $\sim0.6$~meV. In addition, we observe a
distinct splitting of the triplet band of $50(1)$~$\mu$eV or $40(2)$~$\mu$eV at
the band minimum or maximum, respectively. By comparison to a strong coupling
expansion calculation of the triplet dispersion for a spin ladder with
anisotropic exchange, weakly anisotropic leg interactions are identified as the
dominant source of magnetic anisotropy in BPCB. Based on these results, we
discuss the nature of magnetic exchange anisotropy in BPCB and in related
transition-metal insulators.
