  Let $UT_n(F)$ be the algebra of the $n\times n$ upper triangular matrices and
denote $UT_n(F)^{(-)}$ the Lie algebra on the vector space of $UT_n(F)$ with
respect to the usual bracket (commutator), over an infinite field $F$. In this
paper, we give a positive answer to the Specht property for the ideal of the
$\mathbb{Z}_n$-graded identities of $UT_n(F)^{(-)}$ with the canonical grading
when the characteristic $p$ of $F$ is 0 or is larger than $n-1$. Namely we
prove that every ideal of graded identities in the free graded Lie algebra that
contains the graded identities of $UT_n(F)^{(-)}$, is finitely based.
  Moreover we show that if $F$ is an infinite field of characteristic $p=2$
then the $\mathbb{Z}_3$-graded identities of $UT_3^{(-)}(F)$ do not satisfy the
Specht property. More precisely, we construct explicitly an ideal of graded
identities containing that of $UT_3^{(-)}(F)$, and which is not finitely
generated as an ideal of graded identities.
