The four-dimensional rotation group SO(4)
Johan Ernest Mebius∗
April 2008
Abstract
Contribution to Wikipedia (http://en.wikipedia.org/) made in July 2005 as of April 24th 2008
URL http://en.wikipedia.org/wiki/SO%284%29
1
Geometry of 4D rotations
In mathematics SO(4) is the four-dimensional rotation group; that is, the group of rotations about a fixed point in
four-dimensional Euclidean space. The name comes from the fact that it is (isomorphic to) the special orthogonal
group of order 4.
In this article rotation means rotational displacement. For the sake of uniqueness rotation angles are assumed to
be in the segment [0, π] except where mentioned or clearly implied by the context otherwise.
There are two kinds of 4D rotations: simple rotations and double rotations.
1.1
Simple rotations
A simple rotation R about a rotation centre O leaves an entire plane A through O (axis-plane) pointwise invariant.
Every plane B that is completely orthogonal (see Notes) to A intersects A in a certain point P . Each such point
P is the centre of the 2D rotation induced by R in B. All these 2D rotations have the same rotation angle α.
Half-lines from O in the axis-plane A are not displaced; half-lines from O orthogonal to A are displaced through
α; all other half-lines are displaced through an angle < α.
1.2
Double rotations
A double rotation R about a rotation centre O leaves only O invariant. Any double rotation has at least one pair
of completely orthogonal planes A and B through O that are invariant as a whole, i.e. rotated in themselves. In
general the rotation angles α in plane A and β in plane B are different. In that case A and B are the only pair
of invariant planes, and half-lines from O in A, B are displaced through α, β, and half-lines from O not in A or
B are displaced through angles strictly between α and β.
1.3
Isoclinic rotations
If the rotation angles of a double rotation are equal then there are infinitely many invariant planes instead of
just two, and all half-lines from O are displaced through the same angle. Such rotations are called isoclinic
or equiangular rotations, or Clifford displacements. Beware: not all planes through O are invariant under isoclinic rotations; only planes that are spanned by a half-line and the corresponding displaced half-line are invariant.
There are two kinds of isoclinic 4D rotations. To see this, consider an isoclinic rotation R, and take an ordered set OU, OX, OY, OZ of mutually perpendicular half-lines at O (denoted as OU XY Z) such that OU and
OX span an invariant plane, and therefore OY and OZ also span an invariant plane. Now assume that only the
rotation angle α is specified. Then there are in general four isoclinic rotations in planes OU X and OY Z with
rotation angle α, depending on the rotation senses in OU X and OY Z.
We make the convention that the rotation senses from OU to OX and from OY to OZ are reckoned positive. Then we have the four rotations R1 = (+α, +α), R2 = (−α, −α), R3 = (+α, −α) and R4 = (−α, +α). R1
and R2 are each other’s inverses; so are R3 and R4 .
∗ Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, P.O.Box 5031, NL – 2600
GA Delft, The Netherlands, Phone +31.15.2783072, E-Mail jemebius at xs4all dot nl
1
Isoclinic rotations with like signs are denoted as left-isoclinic; those with opposite signs as right-isoclinic.
The four rotations are pairwise different except if α = 0 or α = π. The case α = 0 corresponds to the non-rotation;
α = π corresponds to the central inversion. These two elements of SO(4) are the only ones which are left- and
right-isoclinic.
Left- and right-isocliny defined as above seem to depend on which specific isoclinic rotation was selected. However,
when another isoclinic rotation R0 with its own axes OU 0 X 0 Y 0 Z 0 is selected, then one can always choose the order
of U 0 , X 0 , Y 0 , Z 0 such that OU XY Z can be transformed into OU 0 X 0 Y 0 Z 0 by a rotation rather than by a rotationreflection. Therefore, once one has selected a system OU XY Z of axes that is universally denoted as right-handed,
one can determine the left or right character of a specific isoclinic rotation.
1.4
Group structure of SO(4)
SO(4) is a noncommutative 6-parameter Lie group.
Each plane through the rotation centre O is the axis-plane of a commutative subgroup isomorphic to SO(2). All
these subgroups are mutually conjugate in SO(4).
Each pair of completely orthogonal planes through O is the pair of invariant planes of a commutative subgroup
of SO(4) isomorphic to SO(2) × SO(2).
These groups are maximal tori of SO(4), which are all mutually conjugate in SO(4).
All left-isoclinic rotations form a noncommutative subgroup SL3 of SO(4) which is isomorphic to the multiplicative
3
group S 3 of unit quaternions. All right-isoclinic rotations likewise form a subgroup SR
of SO(4) isomorphic to
3
3
3
S . Both SL and SR are maximal subgroups of SO(4).
Each left-isoclinic rotation commutes with each right-isoclinic rotation. This implies that there exists a direct
3
3
product SL3 × SR
with normal subgroups SL3 and SR
; both of the corresponding factor groups are isomorphic to
the other factor of the direct product, i.e. isomorphic to S 3 .
Each 4D rotation R is in two ways the product of left- and right-isoclinic rotations RL and RR . RL and RR are
together determined up to the central inversion, i.e. when both RL and RR are multiplied by the central inversion
their product is R again.
3
3
are normal subgroups of SO(4).
This implies that SL3 × SR
is the double cover of SO(4) and that SL3 and SR
The non-rotation I and the central inversion −I form a group C2 of order 2, which is the centre of SO(4) and of
3
. The centre of a group is a normal subgroup of that group. The factor group of C2 in SO(4)
both SL3 and SR
3
are isomorphic to SO(3). The factor
is isomorphic to SO(3) × SO(3). The factor groups of C2 in SL3 and SR
3
groups of SL3 and SR
in SO(4) are isomorphic to SO(3).
1.5
Special property of SO(4) among rotation groups in general
The odd-dimensional rotation groups do not contain the central inversion and are simple groups.
The even-dimensional rotation groups do contain the central inversion −I and have the group C2 = {I, −I} as
their centre. From SO(6) onwards they are almost-simple in the sense that the factor groups of their centres are
simple groups.
SO(4) is different: there is no conjugation by any element of SO(4) that transforms left- and right-isoclinic
rotations into each other. Reflections transform a left-isoclinic rotation into a right-isoclinic one by conjugation,
and vice versa. This implies that under the group O(4) of all isometries with fixed point O the subgroups SL3
3
and SR
are mutually conjugate and so are not normal subgroups of O(4). The 5D rotation group SO(5) and all
higher rotation groups contain subgroups isomorphic to O(4). Like SO(4), all even-dimensional rotation groups
contain isoclinic rotations. But unlike SO(4), in SO(6) and all higher even-dimensional rotation groups any pair
of isoclinic rotations through the same angle is conjugate. The sets of all isoclinic rotations are not even subgroups
of SO(2N ), let alone normal subgroups.
2
Algebra of 4D rotations
SO(4) is commonly identified with the group of orientation-preserving isometric linear mappings of a 4D vector
space with inner product over the reals onto itself.
With respect to an orthonormal basis in such a space SO(4) is represented as the group of real 4th-order orthogonal
matrices with determinant +1.
2
2.1
Isoclinic decomposition
A 4D rotation given by its matrix is decomposed into

a00

 a10

A=
 a20

a30
a left-isoclinic and a right-isoclinic rotation as follows: Let

a01 a02 a03

a11 a12 a13 

(1)

a21 a22 a23 

a31 a32 a33
be its matrix with respect to an arbitrary orthonormal basis.
Calculate from this the so-called associate matrix


a00 + a11 + a22 + a33
+a10 − a01 − a32 + a23
+a20 + a31 − a02 − a13
+a30 − a21 + a12 − a03

1
 a10 − a01 + a32 − a23
M= 
4  a20 − a31 − a02 + a13

a30 + a21 − a12 − a03
−a00 − a11 + a22 + a33
+a30 − a21 − a12 + a03
−a30 − a21 − a12 − a03
−a00 + a11 − a22 + a33
+a20 − a31 + a02 − a13
−a10 − a01 − a32 − a23

−a20 − a31 − a02 − a13 

.
+a10 + a01 − a32 − a23 

−a00 + a11 + a22 − a33
(2)
M has rank one and is of unit Euclidean norm as a 16D vector if and only if A is indeed a 4D rotation matrix
([MEBI 2005]). In this case there exist reals a, b, c, d; p, q, r, s such that

 
 ap aq ar as
a
p q r s

 

 bp bq br bs   b 

 

(3)
M =
=

 cp cq cr cs   c 

 

dp dq dr ds
d
and (ap)2 +...+(ds)2 = (a2 +b2 +c2 +d2 )(p2 +q 2 +r2 +s2 ) = 1. There are exactly two sets of a, b, c, d; p, q, r, s
such that a2 + b2 + c2 + d2 = 1 and p2 + q 2 + r2 + s2 = 1. They are each other’s opposites.
The rotation matrix then equals

ap − bq − cr − ds −aq − bp + cs − dr

 bp + aq − dr + cs −bq + ap + ds + cr

A=
 cp + dq + ar − bs −cq + dp − as − br

dp − cq + br + as −dq − cp − bs + ar


a −b −c −d


 b
a −d
c 


=

 c
d
a −b 


d
−c
b
a

−ar − bs − cp + dq
−as + br − cq − dp
−br + as − dp − cq

−bs − ar − dq + cp 


−cs − dr + aq − bp 

−ds + cr + bq + ap
−cr + ds + ap + bq
−dr − cs + bp − aq


p −q −r −s


 q
p
s −r 


.

 r −s
p
q 


s
r
−q
(4)
(5)
p
This formula is due to Van Elfrinkhof ([ELFI 1897]).
The first factor in this decomposition represents a left-isoclinic rotation, the second factor a right-isoclinic rotation.
The factors are determined up to the negative 4th-order identity matrix, i.e. the central inversion.
2.2
Relation to quaternions
A point in 4D space with Cartesian coordinates (u, x, y, z) may be represented by a quaternion u + xi + yj + zk.
A left-isoclinic rotation is represented by left-multiplication by a unit quaternion QL = a + bi + cj + dk.
In matrix-vector language this is
3

u0


a
−b
−c −d
 0  
 x   b
a −d
 

 0 =
 y   c
d
a
 

0
d −c
b
z

u





c 
 x 
.

 y 
−b 


z
a
(6)
Likewise, a right-isoclinic rotation is represented by right-multiplication by a unit quaternion
QR = p + qi + rj + sk, which is in matrix-vector form


 0  
u
p −q −r −s
u


 0  


 x   q
p
s −r 
 x 
 

.

 0 =
 y 
 y   r −s
p
q


 

z0
s
−q
r
(7)
z
p
In the preceding section (Isoclinic decomposition) it is shown how a general 4D rotation is split into left- and
right-isoclinic factors.
In quaternion language Van Elfrinkhof’s formula reads
u0 + x0 i + y 0 j + z 0 k = (a + bi + cj + dk)(u + xi + yj + zk)(p + qi + rj + sk),
or in symbolic form P 0 = QL P QR .
Quaternion multiplication is associative. Therefore P 0 = (QL P )QR = QL (P QR ), which shows that left-isoclinic
and right-isoclinic rotations commute.
2.3
The Euler-Rodrigues formula for 3D rotations
Our ordinary 3D space is conveniently treated as the subspace with coordinate system OXY Z of the 4D space
with coordinate system OU XY Z. Its rotation group is identified with the subgroup of SO(4) consisting of the
matrices


1
0
0
0


 0 a11 a12 a13 


(8)


 0 a21 a22 a23 


0 a31
a32
a33
In Van Elfrinkhof’s formula in the preceding subsection this restriction to three dimensions leads to
p = a, q = −b, r = −c, s = −d, or in quaternion representation: QR = Q0L = Q−1
L ([MEBI 2007]).
The 3D rotation matrix then becomes

  2
a11 a12 a13
a + b2 − c2 − d2

 
 a21 a22 a23  = 
2ad + 2bc

 
a31 a32 a33
−2ac + 2bd
−2ad + 2bc
a2 − b2 + c2 − d2
2ab + 2cd
2ac + 2bd


−2ab + 2cd 

2
2
2
2
a −b −c +d
(9)
with the relation a2 + b2 + c2 + d2 = 1,
which is the representation of the 3D rotation by its Euler-Rodrigues parameters.
The corresponding quaternion formula P 0 = QP Q−1 , where Q = QL and Q−1 = QR , or, in expanded form:
x0 i + y 0 j + z 0 k = (a + bi + cj + dk)(xi + yj + zk)(a − bi − cj − dk) is known as the Hamilton-Cayley formula.
4
See also
Wikipedia lemma
URL
orthogonal matrix
orthogonal group
rotation group
Lorentz group
Poincaré group
Laplace-Runge-Lenz vector
http://en.wikipedia.org/wiki/Orthogonal%5Fmatrix
http://en.wikipedia.org/wiki/Orthogonal%5Fgroup
http://en.wikipedia.org/wiki/Rotation%5Fgroup
http://en.wikipedia.org/wiki/Lorentz%5Fgroup
http://en.wikipedia.org/wiki/Poincar%C3%A9%5Fgroup
http://en.wikipedia.org/wiki/Laplace-Runge-Lenz%5Fvector
Notes
Two flat subspaces S1 and S2 of dimensions M and N of a Euclidean space S of at least M + N dimensions are
called completely orthogonal if every line in S1 is orthogonal to every line in S2 . If dim(S) = M + N then S1 and
S2 intersect in a single point O. If dim(S) < M +N then S1 and S2 may or may not intersect. If dim(S) = M +N
then a line in S1 and a line in S2 may or may not intersect; if they intersect then they intersect in O.
Literature: [SCHO 1902]
References and background literature
ELFR
1897
L. van Elfrinkhof: Eene eigenschap van de orthogonale substitutie van de vierde orde.
Handelingen van het 6e Nederlandsch Natuurkundig en Geneeskundig Congres, Delft, 1897.
http://www.xs4all.nl/%7Ejemebius/Elfrinkhof.htm
(URL not mentioned in original Wikipedia lemma)
KLEI
1932
Felix Klein: Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis.
Translated by E.R. Hedrick and C.A. Noble. The Macmillan Company, New York, 1932.
MANN 1914
Henry Parker Manning: Geometry of four dimensions. The Macmillan Company, 1914.
Republished unaltered and unabridged by Dover Publications in 1954. (∗) (∗∗)
MEBI
2005
Johan E. Mebius: A matrix-based proof of the quaternion representation theorem
for four-dimensional rotations.
http://arxiv.org/abs/math.GM/0501249
MEBI
2007
Johan E. Mebius: Derivation of the Euler-Rodrigues formula
for three-dimensional rotations from the general formula for four-dimensional rotations.
http://arxiv.org/abs/math.GM/0701759
SCHO
1902
P. H. Schoute: Mehrdimensionale Geometrie. Volume 1: Die linearen Räume.
Leipzig: G. J. Göschensche Verlagshandlung (Sammlung Schubert XXXV), 1902
SCHO
1905
P. H. Schoute: Mehrdimensionale Geometrie. Volume 2: Die Polytope.
Leipzig: G. J. Göschensche Verlagshandlung (Sammlung Schubert XXXVI), 1905
http://www-groups.dcs.st-and.ac.uk/%7Ehistory/Mathematicians/Schoute.html
(∗) In this monograph four-dimensional geometry is developed from first principles in a synthetic axiomatic way.
Manning’s work can be considered as a direct extension of the works of Euclid and Hilbert to four dimensions.
(∗∗) http://www.brown.edu/Administration/News%5FBureau/Databases/Encyclopedia/search.php?serial=M0090
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Categories
http://en.wikipedia.org/wiki/Category:Lie%5Fgroups
http://en.wikipedia.org/wiki/Category:Quaternions
http://en.wikipedia.org/wiki/Category:4-dimensional%5Fgeometry
6