  This paper presents a method for computing the Killing form of an isotropic
Lie algebra defined over an arbitrary field based on the Killing form of a
subalgebra containing its anisotropic kernel. This approach allows for
streamlined formulas for many Lie algebras of types E6 and E7 and yields a
unified formula for all Lie algebras of inner type E6, including the
anisotropic ones.
