For straights you should not lift the elbow, the same for hooks

It makes the punch fractionaly slower, wastes energy and telegraphs the punch, alot of boxers lift their elbow but this is from a bad habit rather than trying to lift their elbow.

Most common on peoples jab because they are told to corkscrew it (So the jab is like the cross and lands with a horizontal fist) , but if you are going to do this, do it right at the end of a jab. However the reason for the corkscrew is that the weakest punch (jab), lands harder but it is unnecessary