Computing Lie Bracket Using Properties

Question

When computing a Lie bracket using solely its properties, the following properties are confusing me. Assume $f,g\in C^\infty(\mathbb{R})$:

1. $[X,fY]=X(f)Y+f[X,Y]$
2. $[fX,Y]=f[X,Y]-Y(f)X$
3. $[fX,gY]=fg[X,Y]+f(X(g))Y-g(Y(f))X$

For example, take the vector fields $X=z^2\partial_x+\partial_y$ and $Y=y^2\partial_z$. Calculating the Lie bracket of these 2 vector fields using properties, I get

$\begin{align}[X,Y]&=\left[z^2\partial_x+\partial_y,y^2\partial_z\right]\\ &=\left[z^2\partial_x,y^2\partial_z\right]+\left[\partial_y,y^2\partial_z\right]\\ &=\left(y^2z^2\left[\partial_x,\partial_z\right]+z^2\left(X\left(y^2\right)\right)Y-y^2\left(Y\left(z^2\right)\right)X\right)+\left(X\left(y^2\right)Y+y^2\left[\partial_y,\partial_z\right]\right)\quad\text{[Applying 3. and 1.]}\\ &=\left(z^2\left(2y\right)Y-y^2\left(2y^2z\right)X\right)+\left(\left(2y\right)Y\right).\end{align}$

This is where I get confused because the answer is $\left[z^2\partial_x+\partial_y,y^2\partial_z\right]=2y\partial_z-2y^2z\partial_x$ which I don't get from my computations above. Where did I go wrong? Am I not understanding these properties correctly?

Answer 1

Mixing the concrete fields in your computation with the abstract $X$ and $Y$ is probably causing you confusion. Also don't use square brackets as an extra layer of parenthesis if you're dealing with an operation whose notation also involves square brackets, you're just inviting mistakes. Do it like this:

$$\begin{align} [z^2\partial_x+\partial_y, y^2\partial_z] &= {\color{blue}{[z^2\partial_x,y^2\partial_z]}} + {\color{red}{[\partial_y, y^2\partial_z]}} \\ &= {\color{blue}{z^2y^2[\partial_x,\partial_z] + z^2\partial_x(y^2)\partial_z - y^2\partial_z(z^2)\partial_x}} + {\color{red}{y^2[\partial_y,\partial_z] + \partial_y(y^2)\partial_z}} \\ &= 0 + 0 - 2zy^2\partial_x+0+2y\partial_z \\ &= 2y\partial_z - 2y^2z\partial_x\end{align}$$