Can we split fractional derivative while analyzing fractional differential equations?

Fractional derivatives are generalization to classical integer-order derivatives. The rules which are true for classical derivative need not hold for the fractional derivatives, for example, we cannot simply add the fractional orders $\alpha$ and $\beta$ in ${}_0^{C}\mathrm{D}_t^\alpha {}_0^{C}\mathrm{D}_t^\beta$ to produce the fractional derivative ${}_0^{C}\mathrm{D}_t^{\alpha+\beta}$ of order $\alpha+\beta$, in general. In this article we discuss the details of such compositions and propose the conditions to split a linear fractional differential equation into the systems involving lower order derivatives. Further, we provide some examples, which show that the related results in the literature are sufficient but not necessary conditions.