The M-algebra completes the hierarchy of super-exceptional tangent spaces

The conjectured symmetries of M-theory famously involve (1.) brane-extended super-symmetry (the M-algebra) and (2.) exceptional duality symmetry (the $e_{11}$-algebra); but little attention has been given to their inevitable combination.

In this short note, we highlight (by combining results available in the literature) that the local exceptional duality symmetry (the hyperbolic involutory $k_{1,10}\subset e_{11}$) acts on the M-algebra, through the “brane-rotating symmetry” ${\mathrm{sl}}_{32}$, in a way which extends the known hierarchy of finite-dimensional n-exceptional tangent spaces compatibly beyond the traditional bound of $n\le 7$ all the way to $n=11$.