  We define an $AW^*$-TRO as an off-diagonal corner of an $AW^*$-algebra, and
show that the unit ball of an $AW^*$-TRO has an extreme point. In particular,
the unit ball of an injective operator space has an extreme point, which
answers a question raised in the author's previous work [Journal of Operator
Theory, 76(2) (2016), 219-248] affirmatively. We also show that an $AW^*$-TRO
(respectively, an injective operator space) has an ideal decomposition, that
is, it can be decomposed into the direct sum of a left ideal, a right ideal,
and a two-sided ideal in an $AW^*$-algebra (respectively, an injective
$C^*$-algebra). In particular, we observe that $AW^*$-TRO, hence an injective
operator space, has an algebrization which admits a quasi-identity.
