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It is established in TNG Reunion that the arbiter of succession must choose the two strongest claimants to the chancellor's seat; after that, according to that episode, the two must fight for power.

PICARD: This is not a case of simple mediation. You are asking me to choose the next leader of the Klingon Empire.

K'MPEC: No. By tradition, the two strongest challengers fight for the right of succession. As mediator, only you can designate those challengers.

In TNG Redemption when Gowron is challenged by Duras's illegitimate son Picard must decide whether the challenge is valid.

TORAL: (as Picard is about to put the cloak on Gowron's shoulders) There is one. I will challenge him.

[...]

GOWRON: The illegitimate son of Duras cannot rule the High Council.

K'TAL: The Arbiter will consider his validity. Len'mat.

Picard ultimately decides that the challenge is not valid. Subsequently civil war breaks out between the Duras and Gowron factions. My question is therefore: why? Why not accept the challenge as valid and let Duras's son be easily defeated by Gowron (an adult and an experienced warrior)?

Picard is treading a very fine line here. Although he's well aware that denying the challenge will (in all likelihood) lead to a Klingon civil war, the reality is that the entire reason he was chosen is because he's someone who can interpret the rules of succession without fear or favour.

Ultimately it comes down to the fact that his personal code of ethics and the Federation's Prime Directive won't allow him to use his position as Arbiter to accept an unjustified position even if it benefits the Federation or advances the cause of the side that he thinks should win out. He simply can't, in good conscience, accept Toral's challenge.

The screenplay puts it nicely. Although the (corrupt) Klingon Council is willing to accept a boy as a legitimate challenger, the law and precedent don't and neither is Picard.

Picard (continued): But with due respect to the traditions and laws of this High Council, there is no basis for accepting a petition for leadership from a boy who has fought no battles, shed no blood for his people, earned no honor for himself. Perhaps some day he shall. But not now.

[There are murmurs and reactions from everyone. Picard knows what the reaction to his decision will be... knows the cost... but also knows there is no other way.]

_{^{TNG: Redemption - Original Screenplay}}

Note that although this decision results in war and bloodshed, the long-term result is that a more honourable council eventually takes control. Had Picard allowed the challenge, the boy would have died and Gowron would have become a lame-duck Chancellor surrounded by enemies.