Complex sentence: Alsop became BSO's Music Director Designate in the 2006-2007 concert season, and in the 2007--2008 season she assumed the orchestra's directorship, thus becoming its 12th music director and first female music director.
Split into simpler sentences: Alsop became BSO's Music Director Designate in the 2006-2007 concert season. She assumeed the orchestra's directorship beginning with the 2007--2008 season, becoming its 12th music director and first female music director.