  The process of complexification is used to classify a Lie algebra and
identify its Cartan subalgebra. However, this method does not distinguish
between real forms of a complex Lie algebra, which can differ in signature. In
this paper, we show how Cartan decompositions of a complexified Lie algebra can
be combined with information from the Killing form to identify real forms of a
given Lie algebra. We apply this technique to sl(3,O), a real form of e6 with
signature (52,26), thereby identifying chains of real subalgebras and their
corresponding Cartan subalgebras within e6. Motivated by an explicit
construction of sl(3,O), we then construct an abelian group of order 8 which
acts on the real forms of e6, leading to the identification of 8 particular
copies of the 5 real forms of e6, which can be distinguished by their
relationship to the original copy of sl(3,O).
