Given ideals $I$ and $J$ in a Lie algebra $\mathfrak{g}$ over a commutative ring $K$, the ideal product $IJ$ defined as the image of the composite $$I \otimes_K J \hookrightarrow \mathfrak{g} \otimes_K \mathfrak{g} \xrightarrow{[-, -]} \mathfrak{g} $$ is commutative, but not associative in general. It is easy to see why, because the ideal product is given by $$IJ \equiv \{ \Sigma_{i=0}^n [x_i, y_i] \mid \text{$x_i \in I$, $y_i \in J$, $n \in \mathbb{N}$} \},$$ and $[x,y]=-[y,x]$, but the bracket is not associative in general.

Given ideals $I$ and $J$ in a Lie algebra $\mathfrak{g}$ over a commutative ring $K$, the ideal product $[I,J]$ defined as the image of the composite $$I \otimes_K J \hookrightarrow \mathfrak{g} \otimes_K \mathfrak{g} \xrightarrow{[-, -]} \mathfrak{g}$$ is commutative, but not associative in general. It is easy to see why, because the ideal product is given by $$[I,J] \equiv \{ \Sigma_{i=0}^n [x_i, y_i] \mid \text{$x_i \in I$, $y_i \in J$, $n \in \mathbb{N}$} \},$$ and $[x,y]=-[y,x]$, but the bracket is not associative in general.

Given ideals $I$ and $J$ in a Lie algebra $\mathfrak{g}$ over a commutative ring $K$, the ideal product $IJ$ defined as the image of the composite $$I \otimes_K J \hookrightarrow \mathfrak{g} \otimes_K \mathfrak{g} \xrightarrow{[-, -]} \mathfrak{g} $$ is commutative, but not associative in general. It is easy to see why, because the ideal product is given by $$IJ \equiv \{ \Sigma_{i=0}^n x_i y_i \mid \text{$x_i \in I$, $y_i \in J$, $n \in \mathbb{N}$} \},$$ and $[x,y]=-[y,x]$, but the bracket is not associative in general.

Given ideals $I$ and $J$ in a Lie algebra $\mathfrak{g}$ over a commutative ring $K$, the ideal product $IJ$ defined as the image of the composite $$I \otimes_K J \hookrightarrow \mathfrak{g} \otimes_K \mathfrak{g} \xrightarrow{[-, -]} \mathfrak{g} $$ is commutative, but not associative in general. It is easy to see why, because the ideal product is given by $$IJ \equiv \{ \Sigma_{i=0}^n [x_i, y_i] \mid \text{$x_i \in I$, $y_i \in J$, $n \in \mathbb{N}$} \},$$ and $[x,y]=-[y,x]$, but the bracket is not associative in general.