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 ${\mathfrak{e}}_{11}$-algebra); but little attention has been given to their inevitable combination. In this little note, we highlight (by combining results available in the literature) that the $local$ exceptional duality-symmetry (the hyperbolic involutory ${\mathfrak{k}}_{1,10}\subset {\mathfrak{e}}_{11}$) acts on the M-algebra, through the “brane-rotating symmetry” ${\mathfrak{s}\mathfrak{l}}_{32}$, 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 $n=11$.