  The conjecture of a hidden $E_{10}$ symmetry of M-theory is supported by the
close connection between the dynamics of D=11 supergravity near a spacelike
singularity and a truncation of an one-dimensional $\sigma$-model with $E_{10}$
symmetry where all representations beyond SL(10) level $\ell=3$ are omitted. If
this conjecture is right, higher-level representations should especially
capture the dynamics of further M-theory degrees of freedom. Unfortunately, the
level by level determination of $E_{10}$ commutators which is necessary to
extend the model to higher levels is both an involved and toilsome task that
requires computer aid. In this work, some of the relevant problems are exposed
and algorithmic methods are developed which simplify key steps in the
determination of explicit $E_{10}$ commutators at higher levels. As an
application, we compute the commutator of the level-two six-form with itself.
