In page 2 of Hans Samelson's Notes on Lie Algebras, the text gives an example of Lie algebra $A_L$.
Let $A$ be an algebra over $\mathbb{F}$ (a vector space with an associative multiplication $X\cdot Y$). We make $A$ into a Lie algebra $A_L$ (also called $A$ as Lie algebra) by defining $[X,Y]=X\cdot Y-Y\cdot X$. The Jacobi identity holds; just "multiply out".
Multiplying out the Jacobi identity and using associativity of $\cdot$, we have
$X\cdot Y\cdot Z-X\cdot Z\cdot Y+Y\cdot Z\cdot X-Y\cdot X\cdot Z+Z\cdot X\cdot Y-Z\cdot Y\cdot X=0$.
Assuming only associativity, I can't see why this holds.
The Jacobi identity will give the following:
$$[[x,y],z]+[[y,z],x]+[[z,x],y]=\\(xy-yx)z-z(xy-yx)+(yz-zy)x-x(yz-zy)+(zx-xz)y-y(zx-xz)$$
See if you can multiply now, and use associativity to show that the above expression vanishes.