In this article, we investigate additive properties of the Drazin inverse of elements in rings and algebras over an arbitrary field. The necessary and sufficient condition for the Drazin invertibility of $a-b$ is considered under the condition of $ab=\lambda ba$ in algebras over an arbitrary field. Moreover, we give explicit representations of $(a+b)^D$, as a function of $a,b,a^D$ and $b^D$, whenever $a^3b=ba$ and $b^3a=ab$.