def METHOD_NAME(self, data_lists, convert_QQ, convert_NF):
    r"""
    Compute data lists in Normaliz or ``number_field`` backend format and the internal base ring of the data.
    EXAMPLES::
        sage: # optional - pynormaliz, needs sage.rings.number_field
        sage: p = Polyhedron(vertices=[(0,1/2), (2,0), (4,5/6)],
        ....:                base_ring=AA, backend='normaliz')
        sage: def convert_QQ(ieqs, eqs):
        ....:     return [[1000*x for x in ieq] for ieq in ieqs], \
        ....:            [[1000*x for x in eq] for eq in eqs]
        sage: def convert_NF(ieqs, eqs):
        ....:     return ieqs, eqs
        sage: p._compute_data_lists_and_internal_base_ring(
        ....:     [[[1]], [[1/2]]], convert_QQ, convert_NF)
        (([[1000]], [[500]]), Rational Field)
        sage: p._compute_data_lists_and_internal_base_ring(
        ....:     [[[AA(1)]], [[1/2]]], convert_QQ, convert_NF)
        (([[1000]], [[500]]), Rational Field)
        sage: p._compute_data_lists_and_internal_base_ring(
        ....:     [[[AA(sqrt(2))]], [[1/2]]], convert_QQ, convert_NF)
        ([[[a]], [[1/2]]],
         Number Field in a with defining polynomial y^2 - 2 with a = 1.414213562373095?)
    TESTS::
        sage: K.<a> = QuadraticField(-5)                                            # needs sage.rings.number_field
        sage: p = Polyhedron(base_ring=K,       # indirect doctest  # optional - pynormaliz, needs sage.rings.number_field
        ....:                backend='normaliz',
        ....:                vertices=[(a,1/2), (2,0), (4,5/6)])
        Traceback (most recent call last):
        ...
        ValueError: invalid base ring: Number Field in a ... is not real embedded
    Checks that :trac:`30248` is fixed::
        sage: q = Polyhedron(base_ring=AA,      # indirect doctest  # optional - pynormaliz, needs sage.rings.number_field
        ....:                backend='normaliz',
        ....:                rays=[(0, 0, 1), (0, 1, -1), (1, 0, -1)]); q
        A 3-dimensional polyhedron in AA^3 defined as the convex hull of 1 vertex and 3 rays
        sage: -q                                                    # optional - pynormaliz, needs sage.rings.number_field
        A 3-dimensional polyhedron in AA^3 defined as the convex hull of 1 vertex and 3 rays
    """
    from sage.categories.number_fields import NumberFields
    from sage.rings.real_double import RDF
    if self.base_ring() in (QQ, ZZ):
        internal_base_ring = QQ
        internal_data_lists = convert_QQ(*data_lists)
    else:
        # Allows to re-iterate if K is QQ below when data_lists contain
        # iterators:
        data_lists = [tuple(_) for _ in data_lists]
        internal_data_lists = convert_NF(*data_lists)
        if self.base_ring() in NumberFields():
            if not RDF.has_coerce_map_from(self.base_ring()):
                raise ValueError("invalid base ring: {} is a number field that is not real embedded".format(self.base_ring()))
            internal_base_ring = self.base_ring()
        else:
            K, internal_data_lists, hom = _number_field_elements_from_algebraics_list_of_lists_of_lists(internal_data_lists, embedded=True)
            internal_base_ring = K
            if K is QQ:
                # Compute it with Normaliz, not QNormaliz
                internal_data_lists = convert_QQ(*[ [ [ QQ(x) for x in v ] for v in l]
                                               for l in data_lists ])
    return internal_data_lists, internal_base_ring