+ @staticmethod
+ def trace_inner_product(X,Y):
+ r"""
+ The octonions don't know that the reals are embedded in them,
+ so we have to take the e0 component ourselves.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import OctonionHermitianEJA
+
+ TESTS::
+
+ sage: J = OctonionHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: I = J.one().to_matrix()
+ sage: J.trace_inner_product(I, -I)
+ -2
+
+ """
+ return (X*Y).trace().real().coefficient(0)
+
+
+class AlbertEJA(OctonionHermitianEJA):
+ r"""
+ The Albert algebra is the algebra of three-by-three Hermitian
+ matrices whose entries are octonions.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import AlbertEJA
+
+ EXAMPLES::
+
+ sage: AlbertEJA(field=QQ, orthonormalize=False)
+ Euclidean Jordan algebra of dimension 27 over Rational Field
+ sage: AlbertEJA() # long time
+ Euclidean Jordan algebra of dimension 27 over Algebraic Real Field