The nucleus in a non-associative ring

On the Kantor product, II

“…we consider the remaining terms in(4), that are 2x 2 u({u, {x, u}y} + {u, u{x, y}} − {u, {y, u}x}) + 2xu 2 ({x, {x, u}y} + {x, u{x, y}} − {x, {y, u}x}) − 2xu{u, x}({x, u}y + u{x, y} − {y, u}x) + {x, u}(2{y, u}x 2 u + 2u 2 x{y, x} + 2xyu{u, x}) Since (A, {, }) is a Lie algebra and using Leibniz rule, we obtain 2x 2 u(y{u, {x, u}} + {x, u}{u, y} + u{u, {x, y}} − x{u, {y, u}} − {y, u}{u, x}) + 2xu 2 (y{x, {x, u}} + {x, u}{x, y} + u{x, {x, y}} + {x, y}{x, u} − x{x, {y, u}}) − 2xu{u, x}({x, u}y + u{x, y} − {y, u}x) + {x, u}(2{y, u}x 2 u + 2u 2 x{y, x} + 2xyu{u, x}) + 2x 2 u 2 {x, {y, u}} + 2u{y, u}{x, x 2 u} + 2u 3 x{x, {y, x}} + 2u{y, x}{x, u 2 x} + 2xyu 2 {x, {u, x}} + 2u{u, x}{x, xyu}2x 3 u{u, {y, u}} + 2x{y, u}{u, x 2 u} + 2u 2 x 2 {u, {y, x}} + 2x{y, x}{u, u 2 x} + 2x 2 yu{u, {u, x}} + 2x{u, x}{u, xyu}. • Terms of the form {}{}.…”