Abstract. For an (є; Ґ)-color-commutative associative algebra A with an identity element over a field F of characteristic not 2, and for a color-commutative color-derivation subalgebra D of A, denote by A[D] the associative subalgebra of End(A) generated by A (regarding as operators on A via multiplication) and D. It is proved that, as an associative algebra, A[D] is Ґ-graded simple if and only if A is Ґ-graded D-simple. Suppose A is Ґ-graded D-simple. Then, (a) A[D] is a free left A-module; (b) as a Lie color algebra, the subquotient [A[D];A[D]]=Z(A[D])\[A[D];A[D]] is simple (except one minor case), where Z(A[D]) is the color center of A[D]. The structure of this subquotient is explicitly described.