Recall from [1]}, [2]} that a post-Lie algebra \((\mathfrak {h},[\cdot ,\cdot ]_\mathfrak {h},\rhd )\)  consists of a Lie algebra \((\mathfrak {h},[\cdot ,\cdot ]_\mathfrak {h})\)  and a binary product \(\rhd :\mathfrak {h}\otimes \mathfrak {h}\rightarrow \mathfrak {h}\)  such that
\(x\rhd [y,z]_\mathfrak {h}&=&[x\rhd y,z]_\mathfrak {h}+[y,x\rhd z]_\mathfrak {h},\\([x,y]_\mathfrak {h}+x\rhd y-y\rhd x)\rhd z&=&x\rhd (y\rhd z)-y\rhd (x\rhd z).\) 

Вспомним из [1], [2], что пост-Lie алгебра \((\mathfrak{h}, [\cdot, \cdot]_{\mathfrak{h}}, \rhd)\) состоит из алгебры Ли \((\mathfrak{h}, [\cdot, \cdot]_{\mathfrak{h}})\) и бинарного произведения \(\rhd: \mathfrak{h} \otimes \mathfrak{h} \rightarrow \mathfrak{h}\) таких, что

\(x \rhd [y, z]_{\mathfrak{h}} = [x \rhd y, z]_{\mathfrak{h}} + [y, x \rhd z]_{\mathfrak{h}},\\([x, y]_{\mathfrak{h}} + x \rhd y - y \rhd x) \rhd z = x \rhd (y \rhd z) - y \rhd (x \rhd z).\)