The set E of Levi-Civita connections of left-invariant pseudo-Riemannian Einstein 
metrics on a given semisimple Lie group G always includes D, the Levi-Civita 
connection of the Killing form. When G is noncompact, a still wide-open 1975 
conjecture of Dmitry Alekseevsky, if true, would imply that such left-invariant 
Einstein metrics on G are all inde?nite. This talk explicitly describes several 
connected components of E for the groups G of the SL series, which extends an 
earlier description, obtained in collaboration with Swiatoslaw R. Gal, of the 
component C of E containing D. The picture that emerges is consistent with 
Alekseevsky’s conjecture. The approach focuses on connections rather than 
metrics, which has the advantage of simultaneously generalizing and simplifying 
the algebraic aspect of the question.