  Let ${\mathbb{TL}_n^{\! \mathbb Q}} $ be the rational Temperley-Lieb algebra,
with loop parameter $ 2 $. In the first part of the paper we study the
seminormal idempotents $ E_{ \mathfrak{t}} $ for ${\mathbb{TL}_n^{\! \mathbb
Q}}$ for $ \mathfrak{t} $ running over two-column standard tableaux. Our main
result is here a concrete combinatorial construction of $ E_{\mathfrak{t}} $
using Jones-Wenzl idempotents $ {\mathbf{JW}_{\! k}} $ for ${\mathbb{TL}_k^{\!
\mathbb Q}}$ where $ k \le n $.
  In the second part of the paper we consider the Temperley-Lieb algebra
${\mathbb{TL}_n^{\! {\mathbb F}_p}}$ over the finite field $ {\mathbb F}_p$,
where $ p>2$. The KLR-approach to ${\mathbb{TL}_n^{\! {\mathbb F}_p}}$ gives
rise to an action of a symmetric group $ \mathfrak{S}_m$ on ${\mathbb{TL}_n^{\!
{\mathbb F}_p}}$, for some $ m < n $. We show that the $ E_{ \mathfrak{t}} $'s
from the first part of the paper are simultaneous eigenvectors for the
associated Jucys-Murphy elements for $ \mathfrak{S}_m$. This leads to a
KLR-interpretation of the $p$-Jones-Wenzl idempotent $ ^{p}\!{\mathbf{JW}_{\!
n}} $ for ${\mathbb{TL}_n^{\! {\mathbb F}_p}}$, that was introduced recently by
Burull, Libedinsky and Sentinelli.
