This is pretty interesting, and probably new for many.
Narrated Abu Juhaim Al-Ansari:
The Prophet came from the direction of Bir Jamal. A man met him and greeted him. But he did not return back the greeting till he went to a (mud) wall and smeared his hands and his face with its dust (performed Tayammum) and then returned back the greeting.
The apparent reason is: because As-Salaam is Allah’s Attribute. Prophet (SAW) wouldn’t say it unless he was in the state of wudhu, or tayammum in this case.
There’s another incident similar to this one when Prophet (SAW) didn’t return the greeting because he was in the shower. Read it here.
Question: what do we do? We’re not in the state of wudhu all the time. Should we not return greetings to people in such conditions?
The answer to that (in my opinion) is: it’s the difference of ranks/levels. Prophet (SAW) was on the highest. He didn’t think it right for himself to let Allah’s name come on his tongue in an ‘impure’ (not in wudhu, to be more precise) state. It’s definitely a Sunnah you may follow. But not a compulsion. So let’s not argue about the “fiqh of saying Salaam” when we’re having problems being regular in prayers. :)