Commutative power-associative nilalgebras of dimension 4 and characteristic *2 are shown to be nilpotent and all their isomorphism classes are determined. The long-standing conjecture, originally due to A. A. Albert, that a commutative power-associative nilalgebra of finite dimension over a field is nilpotent, has recently been disproved by Suttles [41, who gave a counterexample of dimension 5. This dimension is generally the best possible, for we show here that every commutative power-associative nilalgebra of dimension 4 over a field of characteristic 4 2 is nilpotent, and we determine the isomorphism classes of all such algebras. The proof is elementary and cannot be significantly simplified by the use of the general results of [1] and [2], which require, moreover, additional assumptions about the characteristic. Throughout, A will denote a commutative power-associative nilalgebra of dimension 4 over a field F of characteristic 4 2. The subspace of A generated by elements u, v, w, ... will be denoted by (u, v, w, ). Since every x E A is nilpotent, the powers of x are linearly independent, so we must have xn = 0 for n > 5; the least n such that xn = 0 for anl x E A is called the nilindex. Now the product of any two elements of A can be written as a linear combination of squares, for xy = ,'4[(x + y)2 _ x 2 _ y 2 ]. Therefore, if the nilindex is 2, then every product vanishes. If it is 5, then A = (x, x2, x3, x4) for any x with x4 A 0. These are trivial cases, in each of which there is, up to isomorphism, a unique, associative, algebra. Only the cases of nilindex 3 and 4 are of interest. 1. Nilindex 3. Linearizing the identity (x2)x 0 yields 2[(xy)z + (yz)x + (zx)y] = 0 for all x, y, z E A. Therefore, denoting right Received by the editors January 2, 1974. AMS (MOS) subject classifications (1970). Primary 17A10.