Formal sums over a Lie algebra

Let $𝔩$ be Lie algebra over $𝕂$ and consider formal sums $𝔩 {𝑧}$ over $𝔩$ . Then we have the alternating bilinear map #m/def/fcalc

[-, -] : 𝔩 {𝑧 1} \times 𝔩 {𝑧 2} \to 𝔩 {𝑧 1, 𝑧 2}

so that

[\sum 𝑚 \in 𝕂 𝑥 𝑚 𝑧 𝑚 1, \sum 𝑛 \in 𝕂 𝑦 𝑛 𝑧 𝑛 2] = \sum 𝑚, 𝑛 \in 𝕂 [𝑥 𝑚, 𝑦 𝑛] 𝑧 𝑚 1 𝑧 𝑛 2

where $𝑥 𝑚, 𝑦 𝑛 \in 𝔩$ for all $𝑚, 𝑛 \in 𝕂$ .

See also