What is wrong with the following formation of a killing form or metric matrix for sl(2,r)?

Question

For sl(2,r) take the generators $m_1\!=\!\left[\begin{array}{cc}1&0\\0&-1\end{array}\right], m_2\!=\!\left[\begin{array}{cc}0&1\\0&0\end{array}\right] ,m_3\!=\!\left[\begin{array}{cc}0&0\\1&0\end{array}\right]$ so nonzero structure cst's are $C_{1,2}^2\!=\!-C_{2,1}^2\!=\!2,\;C_{1,3}^3\!=\!-C_{3,1}^3\!=\!-2,\;C_{2,3}^1\!=\!-C_{3,2}^1\!=\!1$ from which obtain a killing or metric matrix $\;gm\!=\!\left[\begin{array}{cc}8&0&0\\0&0&4\\0&4&0\end{array} \right]$ which is non-singular and according text book(s) cannot be true for sl(2,r).

Answer 1

The Killing form of any semisimple Lie algebra is always non-degenerate (and as stated by @paul_garrett this is Cartan's criterion for semisimplicity). What you remember is probaly that the killing form is (negative) definite only for compact Lie algebras. And the form that you obtain for $\mathfrak{sl}(2,\mathbb R)$ indeed is indefinite (since the determinant is negative).