Product of time ordered product

Let $A$ and $B$ represent two normal product of operators and suppose all operators in $A$ commute with operators in $B$. Then is that true we have $T(A)T(B)=T(AB)$? Where $T$ denote the time ordered product. I proved the above argument should be true, but I want to go furthur, what’s the iff conditions for $T(A)T(B)=T(AB)$ to be true?