A non-trivial question concerning the new teachers’ health insurance board proposed by the legislature but opposed by the Governor is which branch of government it would belong to.
It’s obviously not the judiciary.
And as currently structured, the board cannot be an offshoot of the legislature. A legislature has no power to delegate its statewide lawmaking authority to a group of non-legislators operating outside of the normal lawmaking process — unless it is through the rule-making authority of an executive branch agency.
That leaves the executive branch, which makes sense, as this new board is basically a regulatory agency charged with overseeing the actions of school committees in certain aspects of teacher contract negotiations. However, the legislature does not have the power to designate anyone it chooses as makers of administrative rules that ultimately carry the force of law; according to the principle of separation of powers, this power can only be delegated to a constitutionally recognized executive.
This aspect of separation of powers, fundamental to the structures of our state and Federal governments, is spelled out directly in Article IX Section 5 of the Rhode Island Constitution…

The governor shall, by and with the advice and consent of the senate, appoint all officers of the state whose appointment is not herein otherwise provided for and all members of any board, commission or other state or quasi-public entity which exercises executive power under the laws of this state; but the general assembly may by law vest the appointment of such inferior officers, as they deem proper, in the governor, or within their respective departments in the other general officers, the judiciary or in the heads of departments.