Here is the equation
$$\{x_i,L_j\}=\sum\limits_{k}ϵ_{ijk} x_k.$$
Is this equation generalised for any number of dimensions? In which case, would the following example be correct assuming 4 dimensions?
$$\{x_2, L_1\}=ϵ_{213} x_3 + ϵ_{214} x_4.$$
This, however contradicts Susskind stating that the indices $i$, $j$ and $k$ represent the three directions of space.
In any dimension (not necessarily 3), the angular momentum is a bi-vector $$L^{ij}~=~x^ip^j-x^jp^i,\tag{1}$$ cf. e.g. this Phys.SE post.
Together with the canonical Poisson bracket $$\{x^i,x^j\}~=~0,\quad\{x^i,p_j\}~=~\delta^i_j, \quad \{p_i,p_j\}~=~0, \tag{2}$$ def. (1) implies the following form of OP's sought-for equation $$\{L^{ij},x^k\}~\stackrel{(1)+(2)}{=}~x^j\eta^{ik}-x^i\eta^{jk},\tag{3}$$ where $\eta_{ij}$ are components of a metric tensor.
All the answers and comments above are brilliant and professional, however I probably get your point here, as I am also very new to these topics and notations.
For the summation, it is not about any physical meaning or adding any physical quantity along 3 axes.
The summation is here only to work with the Levi-Civita Symbol ϵ, which is defined to have non-zero value only when i , j , k are all different.
Moreover, as others said, the notation and equation only works in 3D, because the design of ϵ is specificly for the right-hand rule in 3D space.
The example in the same page of the book:
{x2, L1} = ϵ213 * x3
which is from the summation:
{x2, L1} = ϵ211 * x1 + ϵ212 * x2 + ϵ213 * x3 = 0 * x1 + 0 * x2 + (-1) * x3 = -x3
This one powerful equation can describe all the different situations of directions with the aid of summation and the Levi-Civita Symbol ϵ.
This is my first day and first answer in stachexchange and I don't understand how to use the style or latex very well so please forgive me for the bad arrangement.
