Very excellent point by a talented young quran student, Fahad Ali Khan from the IIPC FB group, who pointed out that the word ‘muslim’ is never explicitly connected to an ilah/authority other than Allah. This is somewhat disagreeing with the prevalent view (in the IIPC at least) that ‘muslim laka’ in 2/128 shows that it is possible to be muslim (apparently submissive) to other than Allah.
I think the meaning of muslim is to be an agent of wholeness or soundness. muslim laka, muslim lahu and muslim lillah are levels of that agency. In other words, ibrahim in 2/128 was probably at the highest level.
Contrast this with ibada/worship. It IS possible to worship other than Allah but not to be muslim for other than Allah. I think 3/64 says it best:
Say: O followers of the Book! come to an equitable proposition between us and you that we shall not serve any but Allah and (that) we shall not associate aught with Him, and (that) some of us shall not take others for lords besides Allah; but if they turn back, then say: Bear witness that we are muslims
So here we can see that being muslim involves serving Allah and removing other ‘lords’ besides Allah. There is no laka/to you here. There is only muslim. muslim isnt for any other authority except Allah.