  The determination of the CKM element $V_{cb}$ from inclusive semileptonic
$b\to c \ell \bar\nu$ decays has reached a high precision thanks to a
combination of theoretical and experimental efforts. Aiming towards even higher
precision, we discuss two processes that contaminate the inclusive $V_{cb}$
determination; the $b\to u$ background and the contribution of the tauonic
mode: $b\to c(\tau \to \mu\nu\bar{\nu})\bar{\nu}$. Both of these contributions
are dealt with at the experimental side, using Monte-Carlo methods and momentum
cuts. However, these contributions can be calculated with high precision within
the Heavy-Quark Expansion. In this note, we calculate the theoretical
predictions for these two processes. The $b\to u$ results are compared with
generator-level Monte-Carlo results used at Belle and Belle II. We have good
agreement between theory and Monte-Carlo for lepton energy moments, but less
for hadronic mass moments. Based on our results the uncertainties due to these
backgrounds processes can basically be eliminated by properly including them
into the analyses.
