For two central Drazin invertible elements a and b of a ring, we first prove that a + b is central Drazin invertible under the condition ab = 0. Then we establish the relation between central Drazin invertibility of a + b and 1 + a c b, when a 2 b = aba and b 2 a = bab hold, and also when ab = ba is valid. When a and b are two central group invertible elements, additive properties of central group inverses are studied under the condition ab = ba and also abb © = baa ©.