  In the paper, a new algebra ${\mathcal A}(n)$, which is generated by an upper
triangular generating matrix with triple relations, is introduced. It is shown
that there exists an isomorphism between the algebra ${\mathcal A}(n)$ and the
higher Askey-Wilson algebra ${\mathfrak{aw}}(n)$ introduced by Cramp\'{e},
Frappat et al. Furthermore, we establish a series of automorphisms of
${\mathcal A}(n),$ which satisfy braid group relations and coincide with those
in ${\mathfrak{aw}}(n).$
