CORPORATE PLT TAXIES ONTO ACTIVE RWY.

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THE WRITER TAXIED FROM FBO SE CORNER OF DTW ARPT. THE WRITER-IN-COMMAND WAS ISSUED TAXI CLRNC TO DESIGNATED RWY FOR TKOF RWY 3 AND RWY 21; ACTIVE RWY BEING RWY 3. I MISTAKENLY TAXIED ONTO ACTIVE RWY 3 WITHOUT CLRNC FROM THE TWR. BEING STILL ON GND CTL; I DID NOT REALIZE I HAD TAXIED ONTO AN ACTIVE RWY. THE CIRCUMSTANCES WERE TWOFOLD: THE RWY CTL STRIPES WERE COVERED WITH SNOW AND THERE WAS A CORPORATE JET IN FRONT OF ME BACK-TAXIING ON ACTIVE RWY 3 HDG 210 DEGS AND CROSSED IN FRONT OF ME R TO L. THIS ADDED TO MY ASSUMPTION IT WAS A TAXI RATHER THAN AN ACTIVE RWY. IF I WOULD HAVE LOOKED AT MY CHARTS AHEAD OF TIME; I PROBABLY WOULD HAVE KNOWN THE ERROR OF MY WAYS; BUT NEGLECTED TO DO SO. FORTUNATELY; AT NO TIME WAS THERE DANGER OF RUNNING INTO ANOTHER ACFT DUE TO THE BEFORE MENTIONED CORPORATE JET BACK TAXIING IN FRONT OF ME. THANKING YOU IN ADVANCE FOR YOUR HELP AND CONSIDERATION. I TAKE FLYING SERIOUSLY AND ATTEND FLT SAFETY RECURRENCE SCHOOLING FOR PA31 AT LAKELAND; FL; TWICE A YR AND HAVE FOR MANY YRS. I HAVE LEARNED THE HARD WAY THAT HASTE MAKES WASTE.