  The pure spinor superfield formalism reveals that, in any dimension and with
any amount of supersymmetry, one particular supermultiplet is distinguished
from all others. This "canonical supermultiplet" is equipped with an additional
structure that is not apparent in any component-field formalism: a (homotopy)
commutative algebra structure on the space of fields. The structure is
physically relevant in several ways; it is responsible for the interactions in
ten-dimensional super Yang-Mills theory, as well as crucial to any
first-quantised interpretation. We study the $L_\infty$ algebra structure that
is Koszul dual to this commutative algebra, both in general and in numerous
examples, and prove that it is equivalent to the subalgebra of the Koszul dual
to functions on the space of generalised pure spinors in internal degree
greater than or equal to three. In many examples, the latter is the positive
part of a Borcherds-Kac-Moody superalgebra. Using this result, we can interpret
the canonical multiplet as the homotopy fiber of the map from generalised pure
spinor space to its derived replacement. This generalises and extends work of
Movshev-Schwarz and G\'alvez-Gorbounov-Shaikh-Tonks in the same spirit. We also
comment on some issues with physical interpretations of the canonical
multiplet, which are illustrated by an example related to the complex Cayley
plane, and on possible extensions of our construction, which appear relevant in
an example with symmetry type $G_2 \times A_1$.
