  This is the first in a sequence of articles exploring the relationship
between commutative algebras and $E_\infty$-algebras in characteristic $p$ and
mixed characteristic. In this paper we lay the groundwork by defining a new
class of cohomology operations over $\mathbb F_p$ called cotriple products,
generalising Massey products. We compute the secondary cohomology operations
for a strictly commutative dg-algebra and the obstruction theories these
induce, constructing several counterexamples to characteristic 0 behaviour, one
of which answers a question of Campos, Petersen, Robert-Nicoud and Wierstra. We
construct some families of higher cotriple products and comment on their
behaviour. Finally, we distingush a subclass of cotriple products that we call
higher Steenrod operations and conclude with our main theorem, which says that
$E_\infty$-algebras can be rectified if and only if the higher Steenrod
operations vanish coherently.
