+ r"""
+ Compute the trace inner-product of two real-embeddings.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
+ ....: ComplexHermitianEJA,
+ ....: QuaternionHermitianEJA)
+
+ EXAMPLES::
+
+ This gives the same answer as it would if we computed the trace
+ from the unembedded (original) matrices::
+
+ sage: set_random_seed()
+ sage: J = ComplexHermitianEJA.random_instance()
+ sage: x,y = J.random_elements(2)
+ sage: Xe = x.to_matrix()
+ sage: Ye = y.to_matrix()
+ sage: X = ComplexHermitianEJA.real_unembed(Xe)
+ sage: Y = ComplexHermitianEJA.real_unembed(Ye)
+ sage: expected = (X*Y).trace().real()
+ sage: actual = ComplexHermitianEJA.trace_inner_product(Xe,Ye)
+ sage: actual == expected
+ True
+
+ ::
+
+ sage: set_random_seed()
+ sage: J = QuaternionHermitianEJA.random_instance()
+ sage: x,y = J.random_elements(2)
+ sage: Xe = x.to_matrix()
+ sage: Ye = y.to_matrix()
+ sage: X = QuaternionHermitianEJA.real_unembed(Xe)
+ sage: Y = QuaternionHermitianEJA.real_unembed(Ye)
+ sage: expected = (X*Y).trace().coefficient_tuple()[0]
+ sage: actual = QuaternionHermitianEJA.trace_inner_product(Xe,Ye)
+ sage: actual == expected
+ True
+
+ """