Classification of derivation-simple color algebras related to locally finite derivations

We classify the pairs $(A,D)$ consisting of an $(ϵ,\mathrm{\Gamma })$-olor-commutative associative algebra $A$ with an identity element over an algebraically closed field $F$ of characteristic zero and a finite dimensional subspace $D$ of $(ϵ,\mathrm{\Gamma })$-color-commutative locally finite color-derivations of $A$ such that $A$ is $\mathrm{\Gamma }$-graded $D$-simple and the eigenspaces for elements of $D$ are $\mathrm{\Gamma }$-graded. Such pairs are the important ingredients in constructing some simple Lie color algebras which are in general not finitely-graded. As some applications, using such pairs, we construct new explicit simple Lie color algebras of generalized Witt type, Weyl type.