SYM Paclet Symbol

# KillingForm

 KillingForm[elem1, elem2] gives the Killing form between two elements elemi of a linear Lie algebra. KillingForm[L][baseElem1, baseElem2]gives the Killing form of Lie algebra L between two base elements baseElemi. KillingForm[L, {baseElem1, baseElem2}]gives the Killing form of Lie algebra L between two base elements baseElemi.
• The elements of the linear Lie algebra elemi are represented with matrices.
• The Lie algebra L can be provided either from their structure constants or their infinitesimal generator representation. Depending on that, the resulting base elements baseElemi will be expresses either using the symbolic notation with X[i] denoting ith base element of the Lie algebra or by their infinitesimal generator representation and the symboli notation of the base elements.
• The Killing form of a Lie algebra is the symmetric bilinear form K(X, Y)=Tr(ad X ad Y).
Let the Lie algebra:
 Out[2]=
the following commands have the same effect, they give the killing form between the fist element of the chosen base for the Lie algebra
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Let the linear algebra
the killing form between two elements of this Lie algebra is
 Out[3]=