MLG DEVIATING FOR WX EXCEEDED THE DEVIATION LIMITS SPECIFIED BY ARTCC CTLR USING COMMAND AUTHORITY.

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FLT DEPARTED LAX DEST LAS. LOOP 5 DAG VOR CRESO #1 ARR. TRACKING OUTBND ON THE LAX VOR 041 DEG R WE OBSERVED A MASSIVE TSTM OVER THE SAN GABRIEL MOUNTAINS IN A DIRECT POS WITH OUR INTENDED FLT PATH. WX RADAR WAS ON AND SHOWED AN EXTENSIVE AREA OF HEAVY PRECIP AND TURB TO THE N OF OUR COURSE. WE INFORMED ZLA THAT WE WOULD BE UNABLE TO CONTINUE ON THE 041 DEG R TO DAGGETT VOR BECAUSE OF THE STORM. WE WERE TOLD WE COULD ALTER OUR HDG RIGHT OF COURSE; BUT DO NOT PROCEED E OF THE DAGGETT VOR 215 DEG R. PROCEEDING ON APPROX AN 070 DEG HDG WE INFORMED CENTER THAT WE WOULD NOT BE ABLE TO TRACK INBND ON THE 215 DEG R AND AVOID THE STORM CONDITION; AND WOULD HAVE TO SLIGHTLY PARALLEL THE 215 DEG R TO REMAIN ON THE EDGE OF THE STORM. WE WERE TOLD AGAIN TO NOT FLY E OF THE 215 DEG R UNDER ANY CIRCUMSTANCE. WE REQUESTED A HIGHER ALT AND WERE DENIED THAT REQUEST. WE THEN REQUESTED A RIGHT TURN OF 180-270 DEGS TO CIRCUMNAV THE STORM TO THE W; AGAIN DENIED! WE WERE ONLY TOLD THAT A LEFT TURN WOULD BE PERMITTED. WE INFORMED ATC THAT A LEFT TURN WAS IMPOSSIBLE BECAUSE WE WOULD PLACE OURSELVES INTO THE MAIN INTENSITY OF THE TSTM. OUR EXPLANATION WAS NOT ACCEPTED AND THE LEFT TURN ORDER WAS REPEATED. AN ACR LGT PRECEEDING US TOLD ATC ON OUR FREQ THAT NO ONE CAN GET THROUGH ON THE 215 DEG R. THIS PIREP WAS DISREGARDED BY ATC. WE MADE A SLIGHT LEFT TURN AND JUST SKIRTED THE STORM. ICE TURB ENCOUNTERED. WE TURNED RIGHT AND KEPT THE 215 DEG R OFF SCALE LEFT. I TOLD MY F/O IF ATC INSTRUCTS A FURTHER LEFT TURN TO DECLARE AN EMER.