  The dynamical $p$-forms of torus reductions of maximal supergravity theory
have been shown some time ago to possess remarkable algebraic structures. The
set ("dynamical spectrum") of propagating $p$-forms has been described as a
(truncation of a) real Borcherds superalgebra $\mf{V}_D$ that is characterized
concisely by a Cartan matrix which has been constructed explicitly for each
spacetime dimension $11 \geq D \geq 3.$ In the equations of motion, each
differential form of degree $p$ is the coefficient of a (super-) group
generator, which is itself of degree $p$ for a specific gradation (the
$\mf{V}$-gradation). A slightly milder truncation of the Borcherds superalgebra
enables one to predict also the "spectrum" of the non-dynamical $(D - 1)$ and
$D$-forms. The maximal supergravity $p$-form spectra were reanalyzed more
recently by truncation of the field spectrum of $E_{11}$ to the $p$-forms that
are relevant after reduction from 11 to $D$ dimensions. We show in this paper
how the Borcherds description can be systematically derived from the split
("maximally non compact") real form of $E_{11}$ for $D \geq 1.$ This explains
not only why both structures lead to the same propagating $p$-forms and their
duals for $p\leq (D - 2),$ but also why one obtains the same $(D - 1)$-forms
and "top" $D$-forms. The Borcherds symmetries $\mf{V}_2$ and $\mf{V}_1$ are new
too. We also introduce and use the concept of a presentation of a Lie algebra
that is covariant under a given subalgebra.
