+ EXAMPLES::
+
+ sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
+ ....: [-2, 1, -4, 3],
+ ....: [ 9, 10, 11, 12],
+ ....: [-10, 9, -12, 11] ])
+ sage: _unembed_complex_matrix(A)
+ [ -2*i + 1 -4*i + 3]
+ [ -10*i + 9 -12*i + 11]
+ """
+ n = ZZ(M.nrows())
+ if M.ncols() != n:
+ raise ArgumentError("the matrix 'M' must be square")
+ if not n.mod(2).is_zero():
+ raise ArgumentError("the matrix 'M' must be a complex embedding")
+
+ F = QuadraticField(-1, 'i')
+ i = F.gen()
+
+ # Go top-left to bottom-right (reading order), converting every
+ # 2-by-2 block we see to a single complex element.
+ elements = []
+ for k in xrange(n/2):
+ for j in xrange(n/2):
+ submat = M[2*k:2*k+2,2*j:2*j+2]
+ if submat[0,0] != submat[1,1]:
+ raise ArgumentError('bad real submatrix')
+ if submat[0,1] != -submat[1,0]:
+ raise ArgumentError('bad imag submatrix')
+ z = submat[0,0] + submat[1,0]*i
+ elements.append(z)
+
+ return matrix(F, n/2, elements)
+
+
+def RealSymmetricSimpleEJA(n, field=QQ):
+ """
+ The rank-n simple EJA consisting of real symmetric n-by-n
+ matrices, the usual symmetric Jordan product, and the trace inner
+ product. It has dimension `(n^2 + n)/2` over the reals.
+
+ EXAMPLES::
+
+ sage: J = RealSymmetricSimpleEJA(2)
+ sage: e0, e1, e2 = J.gens()
+ sage: e0*e0
+ e0
+ sage: e1*e1
+ e0 + e2
+ sage: e2*e2
+ e2
+
+ TESTS:
+
+ The degree of this algebra is `(n^2 + n) / 2`::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(1,5).abs()
+ sage: J = RealSymmetricSimpleEJA(n)
+ sage: J.degree() == (n^2 + n)/2
+ True
+
+ """
+ S = _real_symmetric_basis(n, field=field)
+ Qs = _multiplication_table_from_matrix_basis(S)
+
+ return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=n)