SCT Front Line Manager reported observing an aircraft below the MVA; advised Controller working to issue a low altitude alert. Controller did not act as instructed by Front Line Manager.

Narrative

Aircraft X was inbound to SAN from the northwest. A cold front was moving through the airspace producing mid level cloud cover throughout San Diego County. Instrument approaches were in required. As Aircraft X was approaching LUCKI; the Controller issued a southbound turn and descent to 5;000 ft. This was to build some space between the arrival ahead. Once the space was adequate; the Controller turned the aircraft towards VYDDA. I had taken a phone call from RNM informing me they were closing. After taking that phone call; I noticed Aircraft X was descending below 5;000 ft. I alerted the SDWR controller (West Radar) position of the situation and the controller responded with 'Ya I cleared him' I instructed the Controller to issue a Low Altitude Alert and the Controller said 'He's almost out of it'. No action was taken by the Controller and Aircraft X landed WOI (Without Incident]).My main concern is the Controller's actions to this event. His responses to me after alerting him to this situation and not issuing the low altitude alert concern me. Traffic was very light throughout the shift and time on position was not an issue. The Controller simply needed to comply with my direction and take action to correct the error. If the Controller had time to tell me 'He's almost out of it'; and 'Ya I cleared him; the Controller had time to issue the low altitude alert and correct the error.