Lie implies flexible

Statement
Any Lie ring is a flexible ring (i.e., a fact about::flexible magma under the Lie bracket). In other words, the Lie bracket satisfies the following identity:

$$\! x,y],x] = [x,[y,x$$

Related facts

 * Alternating implies flexible
 * Alternative implies flexible
 * Skew-commutative implies flexible
 * Commutative implies flexible

Proof
Given: A Lie ring $$L$$ with Lie bracket $$\! [ \, \ ]$$.

To prove: For all $$x,y \in L$$, we have $$x,y],x] = [x,[y,x$$.

Proof: Since the Lie bracket is alternating, we in particular have that $$[a,b] = -[b,a]$$ for all $$a,b \in L$$. Applying this to $$x = a, y = b$$, we obtain:

$$\! [x,y] = -[y,x]$$

Since the Lie bracket is additive in each coordinate, we can take the Lie bracket with $$x$$ on the left on both sides to obtain:

$$\! [x,[x,y]] = -[x,[y,x]] \ \qquad \ (\dagger)$$

Considering $$a = x, b = [x,y]$$, we get:

$$\! [x,[x,y]] = -[[x,y],x] \ \qquad \ (\dagger\dagger)$$

Combining $$(\dagger)$$ and $$(\dagger\dagger)$$ we obtain:

$$\! -[x,[y,x]] = [[x,y],x]$$

Canceling the minus sign from both sides gives the desired result.