  Under a suitable hypothesis, we construct a full set of pairwise orthogonal
maximal vectors in $V^{\otimes n}$, where $V=V(1)$ is the simple module of
highest weight $1$ for the quantized enveloping algebra
$\mathbf{U}(\mathfrak{sl}_2)$. We give a number of applications, one of which
is an orthogonal basis of the simple modules for the Temperley-Lieb algebra
$\text{TL}_n$. We relate this new orthogonal basis to the standard cellular
basis.
