CAPT INSTRUCTOR OF ACR MLG ACFT INADVERTENTLY TOOK OFF WITHOUT AN ATC CLRNC.

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WHILE IN THE PROCESS OF COMPLETING THE CHKLIST; WE WERE QUESTIONED BY GND CTL IF OUR ACFT HAD A WHITE TAIL LIGHT. MY FO RESPONDED THAT WE DID NOT (OUR STROBE LIGHTS WERE NOT ON). AS WE CONTINUED TAXI AND THE 'BEFORE TKOF CHKLIST;' WE SWITCHED TO BUR TWR FREQ IN THE VICINITY OF A2 AND SHORTLY THEREAFTER RECEIVED CLRNC FOR TKOF ON RWY 15. I ACKNOWLEDGED THE CLRNC. THE 'BEFORE TKOF CHKLIST' WAS COMPLETED TO 'THE LINE' AND I CALLED 'BELOW THE LINE' WHICH COMPLETES THE ENTIRE 'BEFORE TKOF CHKLIST.' MY FO COMPLETED THE 'BELOW THE LINE' ITEMS TO THE POINT OF ME TURNING ON OUR LNDG LIGHTS WHICH I DID AS WE ROUNDED THE CORNER AT THE END OF TAXIWAY A TO TURN ON TO RWY 15. AS WE ROLLED INTO POS ON THE RWY; I MADE A CALL ON THE TWR FREQ THAT WE WERE ROLLING. MY REASON FOR MAKING THIS CALL WAS THAT OUR TKOF CLRNC WAS RECEIVED EARLY AND SOME TIME HAD ELAPSED SINCE RECEIVING THE CLRNC. I WANTED TO RECEIVE CONFIRMATION THAT THE CLRNC WAS STILL VALID. I DISTINCTLY REMEMBER MAKING THE 'ROLLING' CALL BECAUSE I STUMBLED GETTING OUR CALL SIGN OUT; CONFUSING IT WITH OUR PREVIOUS INBOUND FLT NUMBER. THERE WAS NO RESPONSE FROM THE TWR TO THIS CALL. AS WE BEGAN OUR TKOF ROLL; WE BOTH NOTICED AN ACFT ROLLING ON RWY 8-26 THROUGH THE INTXN OF OUR RWY (15) AT A FAIRLY FAST SPD. SEEING THAT HE WOULD NOT INTERFERE WITH OUR TKOF; WE CONTINUED OUR TKOF ROLL. ROTATION AND LIFTOFF WERE NORMAL. JUST AFTER LIFTOFF; WE RECEIVED A CALL FROM BUR TWR THAT WE WERE 'CLRED FOR TKOF; MAINTAIN 4000 FT.' I RESPONDED THAT WE WERE ALREADY AIRBORNE AND HAD BEEN PREVIOUSLY CLRED FOR TKOF. THE TWR CTLR STATED THAT HE DIDN'T THINK HE HAD CLRED US FOR TKOF AND I BRIEFLY STATED THAT HE'D BETTER CHK THE TAPES. WE THEN SWITCHED TO DEP FREQ AND CONTINUED THE DEP. UPON REACHING OUR DEST (SEA); I INITIATED A TELEPHONE CALL TO BUR TWR TO RESOLVE WHAT HAD OCCURRED. THE FO AND I BOTH FEEL CONFIDENT THAT WE HEARD A TKOF CLRNC ISSUED TO US; HOWEVER; BUR TWR RPTED THAT THE TAPE CONFIRMS A 'POS AND HOLD' CLRNC WAS ISSUED AND ACKNOWLEDGED BY US. I HAVE NOT HEARD THE TAPE YET OR SEEN A TRANSCRIPT. THIS IS BOTH PERPLEXING AND SHOCKING! THE ONLY EXPLANATIONS FOR THESE APPARENT CONTRADICTIONS APPEAR TO BE THAT WE EITHER RECEIVED AN UNAUTHORIZED RADIO XMISSION SUCH AS HAS BEEN NOTAMED IN THE SOUTHERN CA AREA; OR THAT WE DID MISUNDERSTAND THE CLRNC. I CANNOT EXPLAIN THE RPTED ACKNOWLEDGEMENT OF 'POS AND HOLD.' BUR TWR ALSO STATED THAT NO 'ROLLING' CALL WAS HEARD ON THE TAPE. THAT DOESN'T SEEM POSSIBLE BECAUSE I AM SO CERTAIN IN MAKING THE CALL. IF THE CLRNC WAS MISUNDERSTOOD AND ACKNOWLEDGED; WE BOTH WOULD HAVE HAD TO BE DISTRACTED AT THE SAME TIME WHICH I DON'T BELIEVE WAS THE CASE. BUR IS ALWAYS A 'HEADS UP' OP AND DEMANDS ATTN AND ONE STRIVES TO KEEP HIS ATTN FOCUSED. I WAS EVEN MORE AWARE OF THIS SINCE I WAS CONDUCTING TRAINING WITH A NEW FO. IF WE DO PROVE TO BE AT FAULT IN THIS OCCURRENCE; PERHAPS I SHOULD RE-EVALUATE THE LOAD I AM UNDER IN THESE TRAINING SITUATIONS. I INTEND TO LEARN FROM THIS EXPERIENCE. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: RPTR STATED THAT THE FAA FSDO HAD FOLLOWED UP WITH AN INVESTIGATION WHICH IS CONCLUDING IN AN OFFICIAL LETTER OF CORRECTION BASED ON THE RPTED AIRMEN COMPLETING A PRESCRIBED AMOUNT OF TRAINING. HE WAS AMAZED TO HEAR THAT THE TAPE RECORDING OF THIS INCIDENT REVEALED THAT HE HAD ACKNOWLEDGED TO THE TWR 'TAXI INTO POS AND HOLD.' HE WOULD STILL LIKE TO HEAR THAT FOR HIMSELF. IN ADDITION TO COMPANY TRAINING HE HAS VOLUNTARILY TAKEN HIMSELF OFF OF THE INSTRUCTOR WORK SINCE HE BELIEVES THAT THIS INCIDENT MAY HAVE NOT HAPPENED IF HE WAS NOT INSTRUCTING A NEW FO WITHIN 1 MONTH AND A FEW HRS IN THE MAKE AND MODEL ACFT. HE BELIEVED THAT IT WAS TOO MUCH TO HANDLE TOO SOON IN THAT ACFT. HE BELIEVES THAT OUR RPTING SVC DOES AN EXCELLENT JOB.