Editing Universal enveloping algebra (section)

===Example: Rotation group SO(3)===
The [[rotation group SO(3)]] is of rank one, and thus has one Casimir operator. It is three-dimensional, and thus the Casimir operator must have order (3&nbsp;−&nbsp;1)&nbsp;=&nbsp;2 i.e. be quadratic. Of course, this is the Lie algebra of <math>A_1.</math> As an elementary exercise, one can compute this directly. Changing notation to <math>e_i=L_i,</math> with <math>L_i</math> belonging to the adjoint rep, a general algebra element is <math>xL_1+yL_2+zL_3</math> and direct computation gives

:<math>\det\left(xL_1+yL_2+zL_3-tI\right)=-t^3-(x^2+y^2+z^2)t</math>

The quadratic term can be read off as <math>\kappa^{ij}=\delta^{ij}</math>, and so the squared [[angular momentum operator]] for the rotation group is that Casimir operator. That is,
:<math>C_{(2)} = L^2 = e_1\otimes e_1 +  e_2\otimes e_2 + e_3\otimes e_3</math>
and explicit computation shows that
:<math>[L^2, e_k]=0</math>
after making use of the [[structure constants]]
:<math>[e_i, e_j]=\varepsilon_{ij}^{\;\;k}e_k</math>