John Gill Commentary 2 Kings 2:13

"He took up also the mantle of Elijah that fell from him, and went back, and stood by the bank of the Jordan." — 2 Kings 2:13 (ASV)

He took up also the mantle of Elijah that fell from him

Which he had now no further need of, and Elisha had, having rent his clothes in two; and this falling into his hands was a token of his succeeding him in his office, and that he should have the double portion of his spirit:

and he went back, and stood by the brook of Jordan ;

at the place where he and Elijah had passed over together.