Did Caligula really make his horse a senator?

Incitatus, favourite horse of Emperor Caligula, allegedly lived in a grand marble house. We are told that Caligula loved him so much that he wanted to make him a consul (lawmaker).

Later writers were keen to demonise Caligula, and he was hardly the most ‘balanced’ of emperors, so it is difficult to know whether he really meant it.