In this note, we investigate algebraic loop structures and inverses of elements of a set of all homomorphisms of Lie algebras with a binary operation derived from a Lie algebra comultiplication. As a symmetry phenomenon, we show that if
l
(
1
)
c
and
r
(
1
)
c
are the left and right inverses of the identity
1
:
L
→
L
on a free graded Lie algebra
L
, respectively, based on the Lie algebra comultiplication
ψ
c
:
L
→
L
⊔
L
, then we have
l
(
1
)
=
l
(
1
)
c
and
r
(
1
)
=
r
(
1
)
c
, where
c
:
L
→
L
⊔
L
is a commutator.