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Example 1.
We use the command SimpleLieAlgebraData to retrieve the structure equations for the rank 3 Lie algebra we initialize this algebra, and we calculate the coroots of several root vectors.
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We obtain the Cartan subalgebra and the positive roots using SimpleLieAlgebraProperties
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Calculate the coroot for the first root .
Calculate the coroot for the last root .
Example 2.
We repeat the calculation the first coroot from Example 1 using the optional calling sequence. The restriction of the Killing form to the Cartan subalgebra is needed.
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Here is the same computation in components.