  Let $\mathfrak{T}$ denote the full Toeplitz algebra on the Bergman space of
the unit ball $\mathbb{B}_n.$ For each subset $G$ of $L^{\infty},$ let
$\mathfrak{CI}(G)$ denote the closed two-sided ideal of $\mathfrak{T}$
generated by all $T_fT_g-T_gT_f$ with $f,g\in G.$ It is known that
$\mathfrak{CI}(C(\bar{\mathbb{B}}_n))=\mathcal{K}$ - the ideal of compact
operators and $\mathfrak{CI}(C(\mathbb{B}_n))=\mathfrak{T}.$ Despite these
``extremal cases'', $\mathfrak{T}$ does contain other non-trivial ideals. This
paper gives a construction of a class of subsets $G$ of $L^{\infty}$ so that
$\mathcal{K}\subsetneq\mathfrak{CI}(G)\subsetneq\mathfrak{T}.$
