Here’s something that Hillary Clinton should care about: Senator Joe Lieberman announced July 3 that if he is defeated in the August 8 Democratic Party primary he will run as an independent to seek his Senate seat.

Why should HRC care? Lieberman is being challenged in Connecticut by Ned Lamont, an antiwar multimillionaire Democrat whose campaign is based almost entirely on his complaint that Lieberman has been a cheerleader for George W. Bush’s war in Iraq. While Lieberman is ahead in the polls, Lamont has narrowed the gap to the point that it is conceivable that Lamont could topple the incumbent. But, as Lieberman said on Monday, that will not keep him out of the race, for he will start to collect the 7500 signatures he needs to run as an independent. Lieberman had to make that decision now; the filing deadline for independent candidates is the day after the Democratic primary. Lieberman could not wait to see what happened in the primary before preparing to run as an independent.

Is this a sign that Lieberman fears he will lose? Maybe not. But it is a sign that Lieberman is not willing to risk losing. And he will have to bear a political cost for crafting this two-track strategy. Lieberman’s announcement will probably not help him among Democratic primary voters. He is essentially saying that if the party choses someone else to be its senatorial nominee, he will work to defeat that candidate. That’s not showing much party loyalty–and it’s possible some Democrats in the Nutmeg State will take exception to his threat.

But back to Hillary. This primary race is–or should be–important to her and other Democrats because it shows how the war can split the party. And that could become the dominant theme of the 2008 race for the Democratic presidential nomination. If the war in Iraq remains a mess a year-and-a-half from now, the Democratic presidential primary will be all about what to do in Iraq. Many Democratic primary voters will be looking for an antiwar, pro-withdrawal candidate (Senator Russ Feingold?) and reluctant to vote for any candidate who has supported the war and stood by it (as has Hillary Clinton). Clinton will certainly have the deepest pockets of any of the candidates–and money usually beats all else (though that didn’t work for Howard Dean in 2004). But if Hillary Clinton is on the wrong side of the war (as far as most Democratic primary voters are concerned), the race will be a bitter and divisive one.

Clinton has not cozied up to Bush the way Lieberman has on the war. She has tried to have it both ways by criticizing the execution of the war but not the mission. Such nuance–or hedging–may get her through the nomination process. But, then again, it might not–if there are enough Democrats PO’ed about the war and her support for it. So the junior senator from New York will be paying close attention to what happens next door in Connecticut. The outcome of this contest may be as important for the future of the Democratic Party as any race in November.