  We consider Sklyanin algebras $S$ with 3 generators, which are quadratic
algebras over a field $\K$ with $3$ generators $x,y,z$ given by $3$ relations
$pxy+qyx+rzz=0$, $pyz+qzy+rxx=0$ and $pzx+qxz+ryy=0$, where $p,q,r\in\K$. This
class of algebras has enjoyed much attention. In particular, using tools from
algebraic geometry Artin, Tate and Van Den Berg \cite{ATV2} showed that if at
least two of the parameters $p$, $q$ and $r$ are non-zero and at least two of
three numbers $p^3$, $q^3$ and $r^3$ are distinct, then $S$ is Artin--Schelter
regular. More specifically, $S$ is Koszul and has the same Hilbert series as
the algebra of commutative polynomials in 3 indeterminates. It has became
commonly accepted that it is impossible to achieve the same objective by purely
algebraic and combinatorial means like the Gr\"obner basis technique. The
authors have previously dispelled this belief. However our previous proof was
no less complicated than the one based on algebraic geometry. It used a
construcion of a Gr\"obner basis in a suitable one-sided module over $S$ and
had quite a number of cases to consider. In this paper we exhibit a linear
substitution after which it becomes possible to determine the leading monomials
of a reduced Gr\"obner basis for the ideal of relations of $S$ itself (without
passing to a module). We also find out explicitly (in terms of parameters)
which Sklyanin algebras are isomorphic. The only drawback of the new technique
is that it fails if the characteristic of the ground field equals 3.
