Here's an exercise from my book (exercise 10, chapter 2.1)
Show that the three-dimensional vector space $V=R^3$ forms an associative algebra with respect to the operation $x\uparrow y=(x_1y_1,x_2y_2,x_3y_3)$. Using the shift operator $S$ defined by $xS=(x_2,x_3,x_1)$, find an expression in terms of $\uparrow$ for the vector product $(x\wedge y)\wedge z$? and deduce that $V$ with the product $x\wedge y$ is a Lie Algebra
What does the author means by the vector product? What do those wedges mean in this context?