If you act loyally, you keep supporting your friends, your principles, your team, or your country — you stand by them, including when it is hard. It is the adverb that belongs with loyal: a dog waiting loyally by the door; people who loyally follow a leader. It can feel warm and personal, or serious and public. It does not mean “according to the law”; that is legally, even though the two words are old cousins and do not mean the same thing now.