The structure of E10 at higher A9 levels - a first algorithmic approach

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.