Additive Property of Drazin Invertibility of Elements

In this article, we investigate additive properties of the Drazin inverse of elements in rings and algebras over an arbitrary field. Under the weakly commutative condition of $ab=\lambda ba$, we show that $a-b$ is Drazin invertible if and only if $a{a}^D(a-b)b{b}^D$ is Drazin invertible. Next, we give explicit representations of $(a+b{)}^D$, as a function of $a,b,a^D$ and $b^D$, under the conditions $a^{3}b=ba$ and $b^{3}a=ab$.