Category of nonassociative algebras

Fix a field $k$ . The category of nonassociative $k$ -algebras has $k$ -algebras with a bilinear multiplication as objects, without requiring associativity, commutativity, a unit, or the Jordan identity. Morphisms are $k$ -linear maps preserving that multiplication. Identities and composition are ordinary map composition.

Which product is being forgotten

An associative, alternative, Lie, or Jordan algebra can be considered here with its own specified bilinear product. Changing from an associative product $xy$ to a commutator $[x,y]$ or to $(xy+yx)/2$ changes the algebra structure; it is not merely forgetting an axiom. The catalogue must record that construction explicitly.

Sedenions

The sedenions fit this broad category despite failing the composition-algebra and alternative-algebra identities. No division or Jordan property follows from membership here.