  We link the recent theory of $L$-algebras to previous notions of Universal
Algebra and Categorical Algebra concerning subtractive varieties, commutators,
multiplicative lattices, and their spectra. We show that the category of
$L$-algebras is subtractive and normal in the sense of Zurab Janelidze, but
neither the category of $L$-algebras nor that of pre-$L$-algebras are Mal'tsev
categories, hence in particular they are not semi-abelian. Therefore
$L$-algebras are a rather peculiar example of an algebraic structure.
