The ability triggers whenever you cast an instant or sorcery spell that targets only Beamsplitter Mage and no other object or player, and only if you control another creature that spell could target. If you cast an instant or sorcery spell that has multiple targets and Beamsplitter Mage is chosen as the target in each instance, Beamsplitter Mage’s ability will trigger.

If you don’t control another creature the spell could target as Beamsplitter Mage’s triggered ability resolves, you don’t copy that spell.

If Beamsplitter Mage leaves the battlefield or becomes an illegal target for the spell that caused its ability to trigger, that spell is still copied.

If you copy a spell, you control the copy. It will resolve before the original spell does.

The copy is created on the stack, so it’s not cast. Abilities that trigger when a player casts a spell (like Beamsplitter Mage’s ability itself) won’t trigger.

If the spell that’s copied is modal (that is, it says “Choose one —” or the like), the copy will have the same mode. A different mode cannot be chosen.

If the spell that’s copied has an X whose value was determined as it was cast (like Selective Snare does), the copy has the same value of X.

The controller of a copy can’t choose to pay any alternative or additional costs for the copy. However, effects based on any alternative or additional costs that were paid for the original spell are copied as though those same costs were paid for the copy.

In some rare cases, the original spell can target another creature but the copy can’t (most likely because an effect modified the characteristics of the original spell on the stack and won’t modify the copy). If the copy can’t target the chosen creature, the copy isn’t created.