  In a recent paper (hep-th/9703084) it was conjectured that the imaginary
simple roots of the Borcherds algebra $g_{II_{9,1}}$ at level 1 are its only
ones. We here propose an independent test of this conjecture, establishing its
validity for all roots of norm $\geq -8$. However, the conjecture fails for
roots of norm -10 and beyond, as we show by computing the simple multiplicities
down to norm -24, which turn out to be remakably small in comparison with the
corresponding $E_{10}$ multiplicities. Our derivation is based on a modified
denominator formula combining the denominator formulas for $E_{10}$ and
$g_{II_{9,1}}$, and provides an efficient method for determining the imaginary
simple roots. In addition, we compute the $E_{10}$ multiplicities of all roots
up to height 231, including levels up to $\ell =6$ and norms -42.
