  We present a translation of the Lambek calculus with brackets and the unit
constant, $\mathbf{Lb}^{\boldsymbol{*}}_{\mathbf{1}}$, into the Lambek calculus
with brackets allowing empty antecedents, but without the unit constant,
$\mathbf{Lb}^{\boldsymbol{*}}$. Using this translation, we extend previously
known results for $\mathbf{Lb}^{\boldsymbol{*}}$ to
$\mathbf{Lb}^{\boldsymbol{*}}_{\mathbf{1}}$: (1) languages generated by
categorial grammars based on the Lambek calculus with brackets are context-free
(Kanazawa 2017); (2) the polynomial-time algorithm for deciding derivability of
bounded depth sequents (Kanovich et al. 2017).
