Liz Cheney's Senate run prompts worries of dividing Republicans

(CBS News) The daughter of former Vice President Dick Cheney wants to follow in her dad's footsteps. Liz Cheney says she's running for the U.S. Senate seat in Wyoming, but the move could split the Republican Party in that state, giving Democrats an opening.

Liz Cheney to challenge Sen. Mike Enzi, R-Wyo.

Cheney announced her plans in a five-minute video and implicitly criticized the man she will try to unseat in next year's Republican primary, Wyoming's senior senator, Mike Enzi.

Liz Cheney has made a name for herself as more than just her father's daughter, and her decision to run for elected office was not unexpected.

However, what is surprising to many is the fact that she'll be challenging a fellow Republican and noted conservative who also happens to be a friend of her family's.

The 46-year-old lawyer was a State Department official under President George W. Bush before leaving to help with the Bush-Cheney re-election effort.

She's remained in the public eye, working for the Republican Party and as commentator on Fox News.

Cheney has been a staunch critic of President Obama and has signaled her political aspirations.

"I think that a life in politics, a career in politics is a really honorable profession, and I have a lot of respect for people who do throw their hat in the ring," she said in a 2009 interview with CBS News' Bob Schieffer.

But her decision to take on a sitting Republican senator means a hard-fought campaign that now is dividing Republicans.

Liz Cheney's Senate ambition causes grief for Wyoming GOP

"It's going to have senator against senator. It's going have the Senate committee that's in charge of re-electing senators against the Cheney family," said Ari Fleisher, who was White House press secretary during the Bush-Cheney years. "It's an uncomfortable split within the Republican Party that could've and should've been avoided."

He now says that it is not the time for Liz Cheney to run.

"The idea of taking on an incumbent in a primary, especially a conservative one like Senator Enzi, I just don't like the timing of it," said Fleisher. "I don't like the intra-Republican Party fight. I wish she had picked another year."

For Jan Crawford's full report, watch the video in the player above.