There is nothing really about any of the types that would contradict being a 4. Imo it is one of the more ubiquitous e-types like 3 or 9.
I think Jack White might be an example of an LII 4, only he is probably sx/sp rather than sp-primary. From his interviews the reactive triad seemed fitting making either 4 or 6 be the most likely choices. There is something very emotionally tortured about his music and lyrics suggestive of longing and melancholy that makes type 4 seem like a better fit for him than a 6.
edit: ahahaha! cracked even has a flowchart for his style