The orientation on the main loadings should be established before the maximisation of the hyperplane count, in order to avoid a definition of factor position only by variables, which do not load on a factor. The hyperplane count is then maximised starting from a position which is optimal with regard to the orientation of the factors on the main loadings. The rotation for orientation on the main loadings is performed successively for every pair of two factors (factor plane). Figure 1 shows the starting position for oblique factor rotation in a factor plane.
Figure 1: Starting position for oblique factor rotation in a factor plane.
It may be seen in Figure 1, that factor F1 should be rotated to a degree which corresponds closely to the mean of the loadings on factor F2 for the variables with main loadings on F1. When this mean is computed, one has to be cautious with the variables with main loadings on F2, but which have low loadings on F1. Those variables should not influence the rotation of F1. Due to the fact that, in most cases, it would be difficult to make a cut-off point for the variables with main loadings on F1, a weighted mean will be computed here. The loadings on F1 are squared, so that the differences of the loadings on F1 will be more pronounced. The weighted mean for the rotation of F1 is computed as follows:
T1 is an element for F1 in the transformation-matrix for the rotation of
the factors. The sign of the loadings on F1 (
)
will not be ignored, when
they are squared (this is realised by computing the third power and by
dividing it by the absolute value of the loading). Because T1 is a weighted
mean, the denominator contains the weights of the numerator. But in this
weighted mean, variables which are factorially complex, i.e. which load
high on the two factors F1 and F2, will be weighted as much as variables
which load high on F1 and low on F2. This may produce oblique rotations
which do not maximise simple structure. In accordance with Hakstian (1971)
a solution should only be very oblique, when variables are factorially
simple. In addition, a solution should be more oblique, only if it becomes
simpler (see Katz and Rohlf, 1975). It can be concluded, that in oblique
rotation, factorial simple variables should be favoured (Warburton, 1963).
This can be realised with an index of factorial simplicity (or complexity). With such an index, the variables may be weighted according to their simplicity. Some indices of factorial simplicity are discussed in Beauducel (1996), e.g. the indices of Hochhausen (1976). The indices were evaluated according to the orientation of factor axes on the main loadings and giving a good starting position for maximisation of the hyperplane count. An index which yielded optimal positions of factor axes was:
The index R is based on the relation between the squared loading (
) on
the factor, which will be rotated, and the absolute loadings on the other
factor (
) of a factor plane. A simple ratio between the absolute
loadings would have the disadvantage, that near-zero loadings on the factor,
which will not be rotated would lead to extreme values. In order to assure
a strong influence of the loadings 
on the index, these loadings are
squared, whereas the loadings 
are not squared. Because one is added in
the denominator, the influence of near-zero loadings on is reduced, and
the index is defined for zero loadings.
When the index R is used as an additional weight of the loadings of F2,
the first non-diagonal element of the transformations-matrix T,
i.e. 
(the element in the second row in the first column) for rotation of F1 is
computed as follows:
The next element of the transformation-matrix is the element for the
rotation of F2 (for the computation only the indices 1 and 2 must be
exchanged in formula 3). Formula 3 will be applied for all factor-planes,
i.e. all non-diagonal elements of the transformation-matrix will be
computed with formula 3, then the transformation-matrix will be normalised
by rows (by dividing each element in the row by the sum of the squared
elements for that row). The matrix of the intermediate solution with
maximised orientation on the main loadings will be called 
.
Conceptually, in this first step of rotation each orthogonal factor is used like a reference vector for every other factor. The simplicity-weighted mean of the orthogonal reference loadings is reduced and the factor will be rotated in the direction of the centroid of the main loadings. Because this transformation is only computed once for every non-diagonal element of the transformation matrix, it leads only to a first departure from the orthogonal solution, which serves as an orientation for the subsequent steps.