  M.-P. Sch\"utzenberger asked to determine the support of the free Lie algebra
${\mathcal L}_{{\mathbb Z}_{m}}(A)$ on a finite alphabet $A$ over the ring
${\mathbb Z}_{m}$ of integers $\bmod m$ and all the corresponding pairs of twin
and anti-twin words, i.e., words that appear with equal (resp. opposite)
coefficients in each Lie polynomial. We study these problems using the adjoint
endomorphism $l^{*}$ of the left normed Lie bracketing $l$ of ${\mathcal
L}_{{\mathbb Z}_{m}}(A)$. Calculating $l^{*}(w)$ via all factors of a given
word $w$ of fixed length and the shuffle product, we recover the result of
Duchamp and Thibon $(1989)$ for the support of the free Lie ring in a much more
natural way. We rephrase these problems, for words of length $n$, in terms of
the action of the left normed multi-linear Lie bracketing $l_{n}$ of ${\mathcal
L}_{{\mathbb Z}_{m}}(A)$ - viewed as an element of the group ring of the
symmetric group ${\mathcal S}_{n}$ - on $\lambda$-tabloids, where $\lambda$ is
a partition of $n$. For words $w$ in two letters, represented by a subset $I$
of $[n] = \{1, 2, ..., n \}$, this leads us to the {\em Pascal descent
polynomial} $p_{n}(I)$, a particular commutative multi-linear polynomial which
equals to a signed binomial coefficient when $|I| = 1$ and allows us to obtain
a sufficient condition on $n$ and $I$ in order that $w$ lies in ${\mathcal
L}_{{\mathbb Z}_{m}}(A)$. We also have a particular conjecture for twin and
anti-twin words for the free Lie ring and show that it is enough to be checked
for $|A| = 2$.
