Cole, for instance, is an ally of House Speaker John Boehner (R-OH) and deputy majority whip, and often acts as a policy trial balloon for Republican leadership. He can express support for potential legislation so that Boehner does not risk the fury of ultra conservative Republicans. Meanwhile Bridenstine is very much in the vein of the tea party and anti-establishment wing of the GOP. He even cast a vote to oust House Speaker John Boehner (R-OH) in 2012.
Coburn's shoes will be tricky to fill. A staunch conservative who nonetheless counts President Barack Obama as a personal friend, Coburn has had a complicated relationship with the tea party -- even derided as a RINO (Republican In Name Only) by some groups despite his hawkish fiscal proposals. Though his now-infamous "bridge to nowhere" speech in 2005 inspired the war against earmarks tea partiers count among their core principles, they staged protests in Oklahoma against the senator.
So, if, say, Cole were to run and win Coburn's seat, it would be held by an establishment Republican who tea partiers revile. If, by contrast, Bridenstine ran and won, the new junior senator from Oklahoma would increase the ranks of anti-establishment conservatives in the Senate like Sen. Ted Cruz (R-TX). Already those outside groups, who have tangled with Republican establishment, sometimes more violently than Democrats, are itching for a fight in Oklahoma.
The powerful Senate Conservatives Fund, long before Coburn announced retirement, endorsed House member Bridenstine as a gesture of encouragement that he might one day run for Senate. The Club for Growth on Friday also indicated that it was not satisfied with the more establishment candidates being mentioned as possible candidates for Coburn's seat.
"We do not know who will run for Senator Coburn's seat, but we do know that Senator Coburn has an outstanding 96 percent lifetime score on the Club for Growth's congressional scorecard and is a champion of economic freedom," Club for Growth spokesman Barney Keller told The Hill. "On the other hand, Congressman Lankford has a lifetime 78 percent and Congressman Cole has a lifetime 73 percent, both of which are obviously substantially different from a score in the 90s."
In other words, they aren't conservative enough for the Club for Growth.
"We'd love to be able to support a candidate that would mirror Senator Coburn's pro-taxpayer record," Keller said.
The seat is regardless a lock for Republicans and Democrats aren't signaling a concerted effort for Coburn's seat. In 2010, The Economist crowned Oklahoma as likely the most conservative state in the country, and Obama lost the state in 2012 by a whopping 33-point margin.
Asked which of Lankford, Cole, or Bridenstine seemed likely to run and possibly win the seat, Oklahoma Republican Party Vice Chairman Sara Jo Odom said she did not know yet.
"Those three names are possibilities, but there are many qualified candidates in Oklahoma," Odom told TPM on Friday. The question is which wing of the GOP will be able to claim Coburn's successor.