-# def projections(self):
-# r"""
-# Return a pair of projections onto this algebra's factors.
-
-# SETUP::
-
-# sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
-# ....: ComplexHermitianEJA,
-# ....: DirectSumEJA)
-
-# EXAMPLES::
-
-# sage: J1 = JordanSpinEJA(2)
-# sage: J2 = ComplexHermitianEJA(2)
-# sage: J = DirectSumEJA(J1,J2)
-# sage: (pi_left, pi_right) = J.projections()
-# sage: J.one().to_vector()
-# (1, 0, 1, 0, 0, 1)
-# sage: pi_left(J.one()).to_vector()
-# (1, 0)
-# sage: pi_right(J.one()).to_vector()
-# (1, 0, 0, 1)
-
-# """
-# (J1,J2) = self.factors()
-# m = J1.dimension()
-# n = J2.dimension()
-# V_basis = self.vector_space().basis()
-# # Need to specify the dimensions explicitly so that we don't
-# # wind up with a zero-by-zero matrix when we want e.g. a
-# # zero-by-two matrix (important for composing things).
-# P1 = matrix(self.base_ring(), m, m+n, V_basis[:m])
-# P2 = matrix(self.base_ring(), n, m+n, V_basis[m:])
-# pi_left = FiniteDimensionalEJAOperator(self,J1,P1)
-# pi_right = FiniteDimensionalEJAOperator(self,J2,P2)
-# return (pi_left, pi_right)
-
-# def inclusions(self):
-# r"""
-# Return the pair of inclusion maps from our factors into us.
-
-# SETUP::
-
-# sage: from mjo.eja.eja_algebra import (random_eja,
-# ....: JordanSpinEJA,
-# ....: RealSymmetricEJA,
-# ....: DirectSumEJA)
-
-# EXAMPLES::
-
-# sage: J1 = JordanSpinEJA(3)
-# sage: J2 = RealSymmetricEJA(2)
-# sage: J = DirectSumEJA(J1,J2)
-# sage: (iota_left, iota_right) = J.inclusions()
-# sage: iota_left(J1.zero()) == J.zero()
-# True
-# sage: iota_right(J2.zero()) == J.zero()
-# True
-# sage: J1.one().to_vector()
-# (1, 0, 0)
-# sage: iota_left(J1.one()).to_vector()
-# (1, 0, 0, 0, 0, 0)
-# sage: J2.one().to_vector()
-# (1, 0, 1)
-# sage: iota_right(J2.one()).to_vector()
-# (0, 0, 0, 1, 0, 1)
-# sage: J.one().to_vector()
-# (1, 0, 0, 1, 0, 1)
-
-# TESTS:
-
-# Composing a projection with the corresponding inclusion should
-# produce the identity map, and mismatching them should produce
-# the zero map::
-
-# sage: set_random_seed()
-# sage: J1 = random_eja()
-# sage: J2 = random_eja()
-# sage: J = DirectSumEJA(J1,J2)
-# sage: (iota_left, iota_right) = J.inclusions()
-# sage: (pi_left, pi_right) = J.projections()
-# sage: pi_left*iota_left == J1.one().operator()
-# True
-# sage: pi_right*iota_right == J2.one().operator()
-# True
-# sage: (pi_left*iota_right).is_zero()
-# True
-# sage: (pi_right*iota_left).is_zero()
-# True
-
-# """
-# (J1,J2) = self.factors()
-# m = J1.dimension()
-# n = J2.dimension()
-# V_basis = self.vector_space().basis()
-# # Need to specify the dimensions explicitly so that we don't
-# # wind up with a zero-by-zero matrix when we want e.g. a
-# # two-by-zero matrix (important for composing things).
-# I1 = matrix.column(self.base_ring(), m, m+n, V_basis[:m])
-# I2 = matrix.column(self.base_ring(), n, m+n, V_basis[m:])
-# iota_left = FiniteDimensionalEJAOperator(J1,self,I1)
-# iota_right = FiniteDimensionalEJAOperator(J2,self,I2)
-# return (iota_left, iota_right)
-
-# def inner_product(self, x, y):
-# r"""
-# The standard Cartesian inner-product.
-
-# We project ``x`` and ``y`` onto our factors, and add up the
-# inner-products from the subalgebras.
-
-# SETUP::
-
-
-# sage: from mjo.eja.eja_algebra import (HadamardEJA,
-# ....: QuaternionHermitianEJA,
-# ....: DirectSumEJA)
-
-# EXAMPLE::
-
-# sage: J1 = HadamardEJA(3,field=QQ)
-# sage: J2 = QuaternionHermitianEJA(2,field=QQ,orthonormalize=False)
-# sage: J = DirectSumEJA(J1,J2)
-# sage: x1 = J1.one()
-# sage: x2 = x1
-# sage: y1 = J2.one()
-# sage: y2 = y1
-# sage: x1.inner_product(x2)
-# 3
-# sage: y1.inner_product(y2)
-# 2
-# sage: J.one().inner_product(J.one())
-# 5
-
-# """
-# (pi_left, pi_right) = self.projections()
-# x1 = pi_left(x)
-# x2 = pi_right(x)
-# y1 = pi_left(y)
-# y2 = pi_right(y)
-
-# return (x1.inner_product(y1) + x2.inner_product(y2))