We're now operating under MGL Ch. 15, Sec. 1F:
Whenever a vacancy occurs in the position of commissioner, the board shall by a two-thirds vote of all its members submit to the secretary, for the secretary's approval, a recommended candidate to fill that vacancy. The secretary may appoint the recommended candidate as commissioner. If the secretary declines to appoint the candidate, the board shall submit a new candidate for consideration. The secretary may appoint the commissioner only from candidates submitted to the secretary by the board.Thus, Secretary Peyser makes the appointment but may do so only from candidates submitted by two-thirds of the Board.
In terms of the Board itself, note that currently there is an open seat, to be filled by Governor Baker, due to the resignation of Roland Fryer. The student representative, who does get a vote, will also join the Board this fall. The Board members, with the exception in this case of Chair Sagan (who has a term coterminus with Governor Baker) serve five year terms, subject to a single renewal.
There's some history in some of the people currently involved around this: more to come on appointing Commissioners.