Semisimple algebra (Killing form)

[droblly]

I have started learning Lie algebra and I can't understand one example given in the notes.
Given:
[h_{\alpha},e_{\alpha}] = 2 e_{\alpha}
[h_{\alpha},f_{\alpha}] = -2 f_{\alpha}
[e_{\alpha},f_{\alpha}] = h_{\alpha}

and that
<br /> [x,y] = K(x,y) t_{\alpha} <br />
if \alpha is a root and x \in L_{\alpha}, y \in L_{-\alpha}
Now, the example is application of the theorem to A_2.
Generators are
h_{\alpha} = E_{11} -E_{22}
h_{\beta} = E_{22} -E_{33}
e_{\alpha} = E_{12}
e_{\beta} = E_{23}
e_{-\alpha} = E_{21}
e_{-\beta} = E_{32}
and Postive roots are {\alpha, \beta, \alpha+\beta}.

I am meant to check that
1.\alpha(h_{\alpha}) = \beta(h_{\beta}) =2
2.\alpha(h_{\beta}) = \beta(h_{\alpha}) =-1

I can't do part (2). Part (1) seems simple:
\alpha(h_{\alpha}) = K(t_{\alpha},h_{\alpha} )= K(t_{\alpha},2\frac{t_{\alpha}}{K(t_{\alpha},t_{\alpha})}) = 2
My problem is with finding t_{\alpha} and t_{\beta} to calculate K(t_{\alpha},t_{\alpha}). How would one go about doing it?
Because
\alpha(h_{\beta}) = K(t_{\alpha},h_{\beta} )= K(t_{\alpha},2\frac{t_{\beta}}{K(t_{\beta},t_{\beta})}) = \frac{2}{K(t_{\beta},t_{\beta})} K(t_{\alpha},t_{\beta})
Thanks.

EDIT: I hope I had posted in the right thread. Should I have posted this in HW help?

 

[droblly]

I have got it. It's just calculating the Cartan integers from the basis.

 

[fresh_42]

See also: https://www.physicsforums.com/insights/lie-algebras-a-walkthrough-the-basics/
and the example at the end.