Can a final element, such as an air-to-open control valve, be shared between a BPCS control loop that is the cause of a scenario, and an SIF that is an IPL for the same scenario?

If you used LOPA to determine if you need a SIF and to identify what SIL is required (if any), then you cannot share or partially share a final element.  So, if the BPCS control loop is the cause (initiating event) and that loop uses the final element in question, then that same final element [...]