+ def inner_product(self, other):
+ """
+ Return the parent algebra's inner product of myself and ``other``.
+
+ EXAMPLES:
+
+ The inner product in the Jordan spin algebra is the usual
+ inner product on `R^n` (this example only works because the
+ basis for the Jordan algebra is the standard basis in `R^n`)::
+
+ sage: J = JordanSpinSimpleEJA(3)
+ sage: x = vector(QQ,[1,2,3])
+ sage: y = vector(QQ,[4,5,6])
+ sage: x.inner_product(y)
+ 32
+ sage: J(x).inner_product(J(y))
+ 32
+
+ The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
+ multiplication is the usual matrix multiplication in `S^n`,
+ so the inner product of the identity matrix with itself
+ should be the `n`::
+
+ sage: J = RealSymmetricSimpleEJA(3)
+ sage: J.one().inner_product(J.one())
+ 3
+
+ Likewise, the inner product on `C^n` is `<X,Y> =
+ Re(trace(X*Y))`, where we must necessarily take the real
+ part because the product of Hermitian matrices may not be
+ Hermitian::
+
+ sage: J = ComplexHermitianSimpleEJA(3)
+ sage: J.one().inner_product(J.one())
+ 3
+
+ TESTS:
+
+ Ensure that we can always compute an inner product, and that
+ it gives us back a real number::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: x.inner_product(y) in RR
+ True
+
+ """
+ P = self.parent()
+ if not other in P:
+ raise TypeError("'other' must live in the same algebra")
+
+ return P.inner_product(self, other)
+
+