Quite often, (especially in Ireland) when we hurt ourselves, get sick or injured were almost bubble wrapped as comforting, given 10 liters of boiled 7up and told to say 6 Hail Marys Every hour.
The main determining factor deciding whether and if you should train when injured or not, will of course be the severity of the injury.
If your legs hanging off and you lost an eye, I think it’s safe to say you better skip the drop-set, however, If it’s something minor like a small break, resistance training can actually help the healing process and prevent muscle atrophy (loss of muscle mass).
There is some pretty decent evidence & research into contralateral training of an immobilized limb aiding in the prevention of muscle atrophy (losing muscle mass) and in some cases even improving strength in the immobilized limb from a process known as Cross Transfer (also known as Cross Education, Contralateral strength Training Effect of Excitatory overflow).
The exact mechanisms of how these works aren’t fully known but it’s hypothesized to be from neural & spinal cord rather than muscular stimulation as there is no direct stimulus of the motor units & muscle fibers of the injured limb. As well as this, cross transfer of circulatory stimulation may help alleviate edema or swelling in the casted limb from better blood flow.
Now, this doesn’t mean load up your a heavy back squat and do some sets of doubles or triples trying to beat your previous best either, submaximal loading is obviously a good idea to just help get some stimulation in the uninsured limb and provide some stimulation of muscle protein synthesis through a contraction.