I think the sibling rivalry is not connected to Don in most cases.

1)Michael - Sonny
No rivalry, they were the closest among brothers.

2)Michael - Tom
This rivalry was on the matter of how much aggressive should one be. Tom meant business, didn't care much about personal insults and did not feel the need to kill everyone. Michael was quite opposite. So michael felt tom was ineffective and incapable. Tom wanted to gain respect but he could never agree with michael's irrational beliefs.

3)Michael - Fredo
This rivalry was based on Fredo's character. Michael considered Fredo as useless and Fredo didn't like it. Only in his own imagination, he was just as capable as anyone, but a victim of others' conspiracy to keep him away from power.

4)Sonny - Fredo
This aspect is not explored much, but I suspect Fredo liked Sonny over michael.

5)Tom - Sonny
Not much rivalry, Sonny did not hate Tom as such and his insults on Tom were not on the same scale as those of michael to tom. Sonny hurt Tom impulsively, in a moment of desaparation - his father was shot, and he was out of his control.

6)Tom - Fredo
This is one place where Don's proximity with Hagen might come into picture. Tom thought Fredo was useless and Fredo didn't care much about Tom's personal opinion but his influence over Don must have bothered Fredo. I am speculating here, we don't get to see much of this being explored in the movie atleast.

7)Connie - brothers
Connie was closest to Sonny and Fredo, she disliked michael(OK, one long post on this is due from me). Fredo and Connie probably were mutually sympathetic because they both did not like michael's attitude. Sonny purely had a protective attitude towards connie (and she didn't really feel bad about it).