In this paper the author considered associative nilpotent algebras over an arbitrary field F, which are not a commutative, but every proper subalgebras and factors are commutative. Such algebras are called C-algebras (by the author). The author investigated the structure of C-algebras R, which satisfies the following conditions: 1) dim R² / R³ = 3;; 2) there is exists an element a of algebra R such as where (k+2) - index of nilpotency of algebra R.

In order to describe C-algebras the author used matrices , which constructed from coefficients of the main relations of this algebras:

The author established the following fact:

Theorem. Let R be 2-generated nilpotent algebra with relations (1) and relation ba = ab + aa^k+1, which also satisfies the condition: dim R^j / R^j+1 = (k - j + 2), for some d. Then R is a C-algebra if and only if for any .