  Let $G$ be a semi-simple simply connected algebraic group over the field
$\mathbb{C}$ of complex numbers. Let $T$ be a maximal torus of $G,$ and let $W$
be the Weyl group of $G$ with respect to $T$. Let $Z(w,\, \underline{i})$ be
the Bott-Samelson-Demazure-Hansen variety corresponding to a tuple
$\underline{i}$ associated to a reduced expression of an element $w \,\in\, W.$
We prove that for the tuple $\underline{i}$ associated to any reduced
expression of a minuscule Weyl group element $w,$ the anti-canonical line
bundle on $Z(w,\,\underline{i})$ is globally generated. As consequence, we
prove that $Z(w,\,\underline{i})$ is weak Fano.
  Assume that $G$ is a simple algebraic group whose type is different from
$A_2.$ Let $S\,=\,\{\alpha_{1},\,\cdots,\,\alpha_{n}\}$ be the set of simple
roots. Let $w$ be such that support of $w$ is equal to $S.$ We prove that
$Z(w,\,\underline{i})$ is Fano for the tuple $\underline{i}$ associated to any
reduced expression of $w$ if and only if $w$ is a Coxeter element and
$w^{-1}(\sum_{t=1}^{n}\alpha_{t})\,\in\, -S$.
