Remark.
For the statement of the theorems we must recall some concepts.
An unramified cyclic cubic relative extension N of a real quadratic field K
with relative automorphism group G = Gal(NK) and unit group U_{N}
is called of principal factorization type (or simply of type) Alpha,
if the cohomology group H^{0}(G,U_{N}) is nontrivial
(i. e., if the unit norm index (U_{K} : Norm_{NK}U_{N}) equals 3)
and it is called of principal factorization type (or simply of type) Delta,
if the cohomology group H^{0}(G,U_{N}) is trivial
(i. e., if the unit norm index (U_{K} : Norm_{NK}U_{N}) equals 1
and thus the fundamental unit of K is norm of a unit of N).

*

Theorem 1. (for the proof see [1])
Let K be a real quadratic field with 3class group of type (3,3).
Suppose that at least three of the four unramified cyclic cubic extensions N_{1},...,N_{4} of K
are of type Alpha, say N_{2},N_{3},N_{4}.
Let the 3class numbers of the nonGalois cubic subfields be
(h(L_{1}),h(L_{2}),h(L_{3}),h(L_{4})) = (3^{u},3,3,3) with u ≥ 1.
Then the metabelian 3group Gal(K_{2}K) of automorphisms of the second Hilbert 3class field K_{2} of K
is of maximal class
and belongs to CF^{2a}(2u+2) with symbolic order Y_{2u}, if N_{1} is of type Delta,
and to CF^{2b}(2u+2) with symbolic order Y_{2u}'
and u ≥ 2, if N_{1} is of type Alpha.
The third possibility that Gal(K_{2}K)
belongs to CF^{a}(2u+1) with symbolic order Y_{2u1}, if N_{1} is of type Alpha,
does obviously not occur (weak leaf conjecture).

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Theorem 2. (for the proof see [1])
Let K be either a real quadratic field with 3class group of type (3,3)
such that all four unramified cyclic cubic extensions N_{1},...,N_{4} of K
are of type Delta,
or a complex quadratic field with 3class group of type (3,3).
Let the 3class numbers of the nonGalois cubic subfields be
(h(L_{1}),h(L_{2}),h(L_{3}),h(L_{4})) = (3^{u},3^{v},3,3) with u ≥ v ≥ 1.
Then the metabelian 3group Gal(K_{2}K) of automorphisms of the second Hilbert 3class field K_{2} of K
is at most of second maximal class
and belongs either to CBF^{a}(2u+2,2u+2v+1) with symbolic order R_{2u,2v}
or to CBF^{b}(2u+3,2u+2v+2).
The isomorphism invariant e has the odd value e = 2v+1.

