HDB1 said:
Dear
@fresh_42 , I am so sorry for bothering you,
please, why you put x1=1, x2=1, x3=1, here, I know it is above, but in general why?
y1, y2, y3 have different values, but xi are the same
$$
K\left(X_1, Y_1\right)+K\left(X_2, Y_2\right)+K\left(X_3, Y_3\right)=4 \cdot 1 \cdot \frac{1}{4}+8 \cdot 1 \cdot \frac{1}{8}+4 \cdot 1 \cdot \frac{1}{4}=3
$$
thanks in advance,
Please correct me if I'm wrong. I assume you refer to post #9. Since we deal with ##3\times 3## matrices, i.e. nine entries, it is better to have a numbered basis, so I set ##X_1=E\, , \,X_2=H\, , \,X_3=F.##
Say we have two arbitrary vectors ##X=x_1X_1+x_2X_2+x_3X_3=x_1E+x_2H+x_3F## and ##Y=y_1X_1+y_2X_2+y_3X_3=y_1E+y_2H+y_3F## then
$$
K(X,Y)=\operatorname{trace}(\operatorname{ad}X \operatorname{ad}Y)=4x_1y_3+8x_2y_2+4x_3y_1\;(*)
$$
This is the general formula obtained by taking the matrices of ##\operatorname{ad}X## and ##\operatorname{ad}Y,## multiplied them, and finally taking the trace.
We need the dual basis of ##\{X_1,X_2,X_3\}## in order to calculate the Casimir element and in particular its trace. Let ##\{Y_1,Y_2,Y_3\}## be that basis. Duality means that ##K(X_i,Y_j)=\delta_{ij}\; (**).##
Hence I wrote the vectors ##Y_j## in the basis ##\{X_1,X_2,X_3\},## i.e. ##Y_j=y_{j1}X_1+y_{j2}X_2+y_{j2}X_3## and solved this linear equation system ##(**)## for the variables ##y_{ij}.## The result was that ##Y_1=(1/4)X_1\, , \,Y_2=(1/8)X_2\, , \,Y_3=(1/4)X_3.##
Finally, I go back to ##(*)## in order to compute ##K(X_i,Y_i).## The formula holds true for any vectors ##X,Y## expressed in the basis ##\{X_1,X_2,X_3\}.## But here we have
\begin{align*}
X_1=x_1X_1+x_2X_2+x_3X_3=1\cdot X_1 +0\cdot X_2+0\cdot X_3& \triangleq (x_1,x_2,x_3)=(1,0,0)\\
X_2=x_1X_1+x_2X_2+x_3X_3=0\cdot X_1 +1\cdot X_2+0\cdot X_3&\triangleq (x_1,x_2,x_3)=(0,1,0)\\
X_3=x_1X_1+x_2X_2+x_3X_3=0\cdot X_1 +0\cdot X_2+1-\cdot X_3&\triangleq (x_1,x_2,x_3)=(0,0,1)\\
Y_1=y_{11}X_1+y_{12}X_2+y_{13}X_3=(1/4)\cdot X_1+0\cdot X_2+0\cdot X_3&\triangleq (y_1,y_2,y_3)=(1/4,0,0)\\
Y_2=y_{21}X_1+y_{22}X_2+y_{23}X_3=0\cdot X_1+(1/8)\cdot X_2+0\cdot X_3&\triangleq (y_1,y_2,y_3)=(0,1/8,0)\\
Y_3=y_{31}X_1+y_{32}X_2+y_{33}X_3=0\cdot X_1+0\cdot X_2+(1/4)\cdot X_3&\triangleq (y_1,y_2,y_3)=(0,0,1/4)
\end{align*}
So our formula ##(*)\; K(X,Y)=4x_1y_3+8x_2y_2+4x_3y_1## becomes
$$
K(X_1,Y_1)=4\cdot 1\cdot (1/4)\, , \,K(X_2,Y_2)=8\cdot 1\cdot (1/8)\, , \,K(X_3,Y_3)=4\cdot 1\cdot (1/4)
$$
and the sum equals ##3,## the dimension of ##\mathfrak{sl}(2).##