We develop a structure theory for nilpotent symplectic alternating algebras.

In this paper and its sequel we continue our study of nilpotent symplectic alternating algebras. In particular we give a full classification of such algebras of dimension 10 over any field. It is known that symplectic alternating algebras over GF(3) correspond to a special rich class [Formula: see text] of 2-Engel 3-groups of exponent 27 and under this correspondence we will see that the nilpotent algebras correspond to a subclass of [Formula: see text] that are those groups in [Formula: see text] that have an extra group theoretical property that we refer to as being powerfully nilpotent and can be described also in the context of [Formula: see text]-groups where [Formula: see text] is an arbitrary prime.

In this paper, we finish our classification of nilpotent symplectic alternating algebras of dimension 10 over any field [Formula: see text].

We continue developing the theory of nilpotent symplectic alternating algebras. The algebras of upper bound nilpotent class, that we call maximal algebras, have been introduced and well studied. In this paper, we continue with the external case problem of algebras of minimal nilpotent class. We show the existence of a subclass of algebras over a field [Formula: see text] that is of certain lower bound class that depends on the dimension only. This suggests a minimal bound for the class of nilpotent algebras of dimension [Formula: see text] of rank [Formula: see text] over any field.