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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 <inline-formula> <math display="inline"> <semantics> <mrow> <mi>l</mi> <msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>c</mi> </msub> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>r</mi> <msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>c</mi> </msub> </mrow> </semantics> </math> </inline-formula> are the left and right inverses of the identity <inline-formula> <math display="inline"> <semantics> <mrow> <mn>1</mn> <mo>:</mo> <mi mathvariant="script">L</mi> <mo>→</mo> <mi mathvariant="script">L</mi> </mrow> </semantics> </math> </inline-formula> on a free graded Lie algebra <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">L</mi> </semantics> </math> </inline-formula>, respectively, based on the Lie algebra comultiplication <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>ψ</mi> <mi>c</mi> </msub> <mo>:</mo> <mi mathvariant="script">L</mi> <mo>→</mo> <mi mathvariant="script">L</mi> <mo>⊔</mo> <mi mathvariant="script">L</mi> </mrow> </semantics> </math> </inline-formula>, then we have <inline-formula> <math display="inline"> <semantics> <mrow> <mi>l</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mi>l</mi> <msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>c</mi> </msub> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>r</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mi>r</mi> <msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>c</mi> </msub> </mrow> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>c</mi> <mo>:</mo> <mi mathvariant="script">L</mi> <mo>→</mo> <mi mathvariant="script">L</mi> <mo>⊔</mo> <mi mathvariant="script">L</mi> </mrow> </semantics> </math> </inline-formula> is a commutator.