E11 as E10 representation at low levels

The Lorentzian Kac–Moody algebra E₁₁, obtained by doubly overextending the compact E₈, is decomposed into representations of its canonical hyperbolic E₁₀ subalgebra. Whereas the appearing representations at levels 0 and 1 are known on general grounds, higher level representations can currently only be obtained by recursive methods. We present the results of such an analysis up to height 120 in E₁₁ which comprises representations on the first five levels. The algorithms used are a combination of Weyl orbit methods and standard methods based on the Peterson and Freudenthal formulae. In the appendices we give all multiplicities of E₁₀ occurring up to height 340 and for E₁₁ up to height 240.