Lie Algebra

f : L × L → L, Cross product f(x, y) = x × y. x × (y × z) + z × (x × y) + y (z × x) = 0 Lie bracket [a x + b y, z] = a [x, z] + b [y, z], [x, a y + b z] = a [x, y] + b [x, z] (bilinear). [x, x] = 0 (alternating). [x, [y, z]] + [z, [x, y]] + [y, [z, x]] = 0 (Jacobi identity). 0 = [x + y, x + y] = [x, x + y] + [y, x + y] = [x, x] + [x, y] + [y, x] + [y, y] = [x, y] + [y, x]
⇒ [x, y] = -[y, x] (anticommutative). Derivation D [x, y] = [D x, y] + [x, D y].

Jacobi Identity

x × (y × z) + z × (x × y) + y (z × x) = 0.

References

• Lie Algebras and Homotopy Theory. Jacob Lurie. 2019.