There were two political parties that held fateful conventions this week. Neither party is known for being particularly democratic. Both have leaders in turmoil who wish they were as popular as their predecessors. And the recovery - or at least stability - of both parties is important to Prime Minister Binyamin Netanyahu. But Labor Party leader Ehud Barak and Fatah chairman Mahmoud Abbas convened their conventions for very different reasons. The fact that Fatah's took place for the first time in 20 years shows that Abbas believes his hold over his party is sufficiently secure, while Barak only used his own convention to consolidate his power, and ensure that he would remain Labor's head for years to come. Barak succeeded in keeping Labor united, at least on paper. The party passed a new constitution that gives him unprecedented powers in the party. While past Labor leaders had to answer to a secretary-general and strong internal courts, Barak will now be unencumbered to run the party on his own, however he likes. The most significant change made at Labor's convention was the repealing of a clause in the party's constitution requiring a leadership race 14 months after every loss in a general election - a clause that resulted in Labor's having six leaders in an eight-year period. Now leadership races will be held ahead of general elections, just like in Likud and Kadima. The end result of the change is that Netanyahu can now rely on Barak's staying at Labor's helm until at least October 2012, instead of April 2010, according to the old constitution. Those two and a half years of political quiet on the left flank of his coalition were a blessing for Netanyahu, who knew that if a Labor rebel MK replaced Barak a mere eight months from now, his government's days could be numbered. Barak can now continue to serve as Netanyahu's chief defender internationally, the head of the negotiating team with the United States, the de facto foreign minister and, most importantly for Netanyahu, his partner in dealing with Iran. BUT, WHILE the left flank of Netanyahu's coalition was strengthened, the Right took a significant hit this week, when the police announced its recommendation to indict Israel Beiteinu head Avigdor Lieberman on corruption charges. Lieberman promised to quit the cabinet, the Knesset and the leadership of his party to fight the charges following an indictment that is expected between September and December. Even though Israel Beiteinu will remain in the coalition when Lieberman leaves, Netanyahu may no longer enjoy the pleasure of having 15 guaranteed hands in favor of whatever he wants to do. Unlike Labor, in which all 13 MKs have their own agendas, Israel Beiteinu - like Shas - is conveniently monolithic. It is possible that Israel Beiteinu MKs will enjoy their newfound independence and split up as fast as Shinui did following the departures of leaders Yosef Lapid and Avraham Poraz. But Netanyahu's associates predict that the opposite will happen, and that Israel Beiteinu's breakup would only make the coalition stronger. Its MKs would continue to fear Lieberman, who could still return and decide their fate, and if he doesn't, they might want to join Likud, which would require loyalty. No matter what happens with Lieberman, Israel Beiteinu MKs would want the next election to be held as late as possible, because their remaining in the Knesset is far from guaranteed. This is also true of Labor MKs - following a Smith Research poll published this week that found that Labor would receive only six seats if elections were held today - and also Shas, which could shrink, due to Arye Deri's new party. That is a recipe for stability. SO, AFTER a week in which Netanyahu's main coalition partner on the Left was strengthened and his main partner on the Right was weakened, the prime minister has emerged stronger politically. In private conversations, Netanyahu's associates admit that even though Lieberman controls 15 MKs and Barak at best influences Agriculture Minister Shalom Simhon and himself, Barak is more valuable, because while Lieberman's replacement as foreign minister would undoubtedly improve Israel's image abroad, Barak is a lot harder to replace. There are plenty of wannabe defense ministers in Likud, but Netanyahu needs someone who can also serve as his bullet-proof vest to the world. The only MK other than Barak capable of performing both functions is Shaul Mofaz, and his attacks on Netanyahu over the past two weeks made his joining the government increasingly unlikely. "Bibi wants Barak, and he trusts him as defense minister," a source close to Netanyahu said. "We feel bad about Ivet [Lieberman's nickname], and we hope the attorney general decides differently than expected, but the government is not unstable without him." Following the ups and downs in Labor and Israel Beiteinu - and also battles this week in Habayit Hayehudi - Netanyahu is looking ahead to one more party whose internal problems could impact the longevity of his premiership: Abbas's Fatah. Netanyahu needs there to be an attractive leadership in Fatah that could prevent the rise of Hamas in the January Palestinian Authority election if it indeed takes place. If Hamas wins, the political ramifications in Israel could be just as significant as the security impact. So even if all the problems in Labor and Israel Beiteinu get resolved in a way that helps the prime minister, one party could still mess everything up.