Complex simple Lie algebras are classified by their root types. The type of a real simple Lie algebra is the root type of the associated complex algebra. The character of a real simple Lie algebra is the signature of its Killing form.
For many root types, the character is sufficient to uniquely classify the corresponding real Lie algebras. However, one should not take this statement to be literally true – there are a few cases where the character does not suffice to distinguish all possible real forms.
In this worksheet we will show that the 2 real non-isomorphic Lie algebras so*(18) and so(12, 6) (with isomorphic complexifications so(18)) have the same character.
Anderson, Ian M., "Non-Isomorphic Real Simple Lie Algebras of the Same Complex Type and Character" (2015). Tutorials on... in 1 hour or less. Paper 6.
Additional FilesNonIsomorphicRealLieAlgebrasOfTheSameCharacter.mw (626 kB)
NonIsomorphicRealLieAlgebrasOfTheSameCharacter.pdf (336 kB)