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In this paper we investigate derivations of a commutative powerassociative algebra. Particular cases of stable and partially stable algebras are inspected. Some attention is paid to the Jordan case. Further results are given. Especially, we show that the core of a n th -order Bernstein algebra which is power-associative is a Jordan algebra.

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In this paper we study the class of algebras satisfying a polynomial identity of degree six that are principal train algebras of rank $3$ or $4$, for which we give the explicit form of the train equation. If the rank of $A$ is $n\geq 5$ in general, we provide the form of the train equation in some cases.