Let TL(n)(Q )be the rational Temperley-Lieb algebra, with loop parameter 2. In the first part of the paper we study the seminormal idempotents E-t for TLnQ for t running over two-column standard tableaux. Our main result is here a concrete combinatorial construction of E t using Jones-Wenzl idempotents JW(k )for TLkQ where k <= n . In the second part of the paper we consider the TemperleyLieb algebra TLn Fp over the finite field F-p , where p > 2. The KLR-approach to TLnFp gives rise to an action of a symmetric group S-m on TLnFp , for some m < n . We show that the E t 's from the first part of the paper are simultaneous eigenvectors for the associated Jucys-Murphy elements for S-m . This leads to a KLR-interpretation of the p-Jones-Wenzl idempotent p JW(n) for TLnFp , that was introduced recently by Burull, Libedinsky and Sentinelli. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.