A B737-NG PILOT REPORTS RESPONDING TO A RA AT 1;500' AFTER A BFI RWY 13 DEP. POOR CRM LEAD TO A DELAYED RESPONSE TO AN ATC DIRECTED TURN.

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THE L SEAT PLT IS A CAPT IN TRAINING AND THE R SEAT PLT IS THE TRAINING PLT. THE L SEAT PLT PRE-BRIEFED THE PROX OF SEA TO BFI PRIOR TO TKOF. THE R SEAT PLT WAS FLYING THE ACFT. THE R SEAT PLT ATTEMPTED TO ENGAGE THE AUTOPLT AT APPROX 400 FT AGL AS HIS BELIEF THAT AUTOMATION SHOULD BE USED ASAP AFTER TKOF. THE AUTOPLT FAILED TO ENGAGE AND THE AUTOPLT AURAL WARNING SOUNDED. I WAS FLYING IN THE L SEAT AND CHANGED MY OBSERVATION TO OUT OF THE L FORWARD AND SIDE WINDOW. NEXT; I HEARD THE CALL 'HDG SELECT' TO WHICH I RESPONDED BY LOOKING TOWARD THE MCP AND THEN ACCIDENTALLY PUSHED THE UNLIT APPR BUTTON (LOCATED NEXT TO THE HDG BUTTON). I IMMEDIATELY PUSHED APPR AGAIN TO DISARM THE APCH MODE AND NOTICED THAT HDG SELECT WAS ALREADY ACTIVE. THE R SEAT PLT STATED 'WHAT ARE YOU DOING; YOU ARE GIVING ME MODES I AM NOT ASKING FOR.' AT THIS POINT ATC GAVE US A L TURN TO 020 DEG HDG. I BEGAN TO TURN THE HDG SELECT TO THE L WHEN THE R SEAT PLT SLAPPED MY HAND AND STATED THE MCP WAS HIS AREA OF RESPONSIBILITY SINCE THE AUTOPLT WAS IN COMMAND (A CHANGE THAT HE SHOULD HAVE VERBALLY CALLED WHEN THE AUTOPLT ENGAGED BUT HE DID NOT MAKE THAT CALL). I WAS OCCUPIED LOOKING OUT THE WINDOW FOR TFC AND DID NOT NOTICE THAT THE AUTOPLT WAS IN COMMAND BECAUSE IN MY PERIPHERAL VIEW; THE R SEAT PLT WAS POSITIONED AT THE CTLS AS IF HE WAS ACTUALLY FLYING AND NO AUTOPLT CHANGE MODES CALLS WERE MADE. AT THIS POINT; HE TOOK OVER CTL OF THE MCP AND HE BEGAN TURNING THE HDG SELECT TO THE R (TOWARD SEATAC). I IMMEDIATELY VERBALLY CORRECTED HIM; TELLING HIM THAT 'THE TURN WAS TO THE L.' HE REQUESTED THAT I CONFIRM THE TURN FROM ATC. IT WAS AT THIS POINT WE RECEIVED THE RA.