On identities of model algebras

A sharp upper bound for the nilpotency index of the commutator ideal of a $2$-generated subalgebra of an arbitrary model algebra is given; this estimate is about half that for arbitrary Lie nilpotent algebras of the same class. All identities in two variables that hold in the model algebra of multiplicity $3$ are found. For any $m\geqslant 3$, in a free Lie nilpotent algebra $F^{(2m+1)}$ of class $2m$, the kernel polynomial of smallest possible degree is indicated. It is proved that the degree of any identity of a model algebra is greater than its multiplicity.