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The following figure is from an article about Casimir element of Wikipedia. According to the following, $L_x^2$ is not a simple matrix multiplication of $L_x$ with itself, since the sign is opposite. It is a multiplication in the universal enveloping algebra. But exactly what caused this sign change?

enter image description here

Math.StackExchange
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  • There's a sign change that might occur coming from the fact that the Killing form is negative rather than positive definite. But I just don't agree with this calculation as written; if you write $L_x^2$ then you should mean $L_x^2$. – Qiaochu Yuan Jan 24 '16 at 17:01
  • I came up with this question, since I thought there would be some convention such that $L^2_x$ is defined as $L^t_xL_x$, which means multiplication of the transposed matrix and the original one. Otherwise the identity doesn't seem to hold. Likewise, if I don't assume the above convention, Casimir element of $\mathcal{so(3)}$ becomes $-\frac{1}{2}(L^2_x+L^2_y+L^2_z)$, which has an incorrect sign. – Math.StackExchange Jan 24 '16 at 17:07

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In such cases it might help to visit the talk page as well, because other users might have run into the same issue.

There is a discussion about this issue here.

mvw
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