What could be the cause of the low beam lights not working when switching from high beam?

The simple answer may be that the low beam side is burnt out. Replace the bulb.

However, if both headlights have stopped working when switching to low beams, there may be a problem with the relay or the switch itself. If the cost of a relay is low for your car, it may be less expensive to simply replace that component than to have it diagnosed in a repair shop. Replacement of the switch requires some disassembly of the steering column or dash in most vehicles. Check a specific service manual.