In the following text quotations from Evans' book manuscript [1] appear in black with equation labels (1.nn) at the right margin.

The claim of O(3)-symmetry is a central concern of Evans' considerations since 1992. The reader will find a historical overview in [2; Sect. 5] written by A. Lakhtakia. Among a lot of papers Evans has written five books on "The Enigmatic Photon" that deal with the claimed O(3)-symmetry of electromagnetic fields.

In [1; Chap.1.2] Evans considers a circularly polarized plane electromagnetic wave propagating in z-direction. Using the electromagnetic phase

Φ = ω t − κ z ,                                                                 (1.38)

where κ = ω/c, Evans describes the wave relative to his complex circular basis [1; (1.41)], see also [3; Appendix 1] and [4]. The magnetic field is given by

B⁽¹⁾ = B⁽⁰⁾q'⁽¹⁾ = ¹/_sqr(2) B⁽⁰⁾ (i−ij) e^iΦ ,                                           
B⁽²⁾ = B⁽⁰⁾ q'⁽¹⁾ = ¹/_sqr(2) B⁽⁰⁾ (i+ij) e^−iΦ ,                               (1.43)
B⁽³⁾ = B⁽⁰⁾ q'⁽³⁾ = B⁽⁰⁾ k ,                                                                

satisfying Evans' "cyclic O(3)-symmetry relations"

B⁽¹⁾ × B⁽²⁾ = iB⁽⁰⁾B⁽³⁾* ,                                                                  
B⁽²⁾ × B⁽³⁾ = iB⁽⁰⁾B⁽¹⁾* ,                                                         (1.44)
B⁽³⁾ × B⁽¹⁾ = iB⁽⁰⁾B⁽²⁾* .                                                                  

Due to M.W. Evans the corresponding electric field is given by

E⁽¹⁾ = − ^E(0)/_sqr(2) (ii+j) e^iΦ ,                                                          

E⁽²⁾ = ^E(0)/_sqr(2) (ii−j) e^−iΦ ,                                                   (1.85)

E⁽³⁾ = −iE⁽⁰⁾ k .                                                                             

The relation between E⁽⁰⁾ and B⁽⁰⁾ is

E⁽⁰⁾ = c B⁽⁰⁾ ,                                                                 (1.87)

i.e. E⁽⁰⁾ = B⁽⁰⁾, if we assume c=1 below.

Due to B⁽²⁾ = B⁽¹⁾* and E⁽²⁾ = E⁽¹⁾* the complex fields (1.43) and (1.85) belong to the real fields

                                B = B⁽¹⁾ + B⁽²⁾ + B⁽³⁾ = B_x i + B_y j + B_z k
and
                                E = E⁽¹⁾ + E⁽²⁾ + E⁽³⁾= E_x i + E_y j + E_z k .

Inserting of (1.43) and (1.85) and coefficient matching yields

(1)                 B_x = B⁽⁰⁾ sqr(2) cos Φ ,         B_y = B⁽⁰⁾ sqr(2) sin Φ ,             B_z = B⁽⁰⁾ ,
(2)                 E_x = E⁽⁰⁾ sqr(2) sin Φ ,         E_y = − E⁽⁰⁾ sqr(2) cos Φ ,         E_z = E⁽⁰⁾ .

Summing the equations in (1) with combination factors 1,+i and comparing with (1.43) yields

(3)                 (B_x + i B_y) (i−ij)= 2 B⁽¹⁾,         (B_x − i B_y) (i+ij) = 2 B⁽²⁾ .

and therefore for further use in rewriting of the first equation of (1.44)

(4)                 B⁽¹⁾ × B⁽²⁾ = ½ (B_x + i B_y) (B_x − i B_y) i k = ½ (B_x˛ + B_y˛) i k .

while the last equations of (1.43) and (1) yield

B⁽⁰⁾B⁽³⁾* = ik B⁽⁰⁾˛ = ik B_z˛ .

Thus, one of "Evans' cyclic symmetry relations", the first rule of (1.44), is equivalent to

(5)                                                                 ½ (B_x˛ + B_y˛) = B_z˛ .

The first two equations of (1.43) and (1.85) describe a circularly polarized plane wave propagating in z-direction. The third equations, however, contain Evans' O(3)-Law from 1992, saying that the well-known plane wave is always accompanied by a constant longitudinal magnetic "ghost" field B⁽³⁾, the size of which - this is important - is given by the third equation of (1.43), or by the first equation of (1.44), which is in real formulation our equation (5).

If Evans' O(3)-Law were a Law of Physics then it must be invariant under the admissible coordinate transforms, i.e. under Lorentz transforms.

Therefore we consider the wave as observed from other coordinate systems S' in constant motion v = v k relative to our original Cartesian coordinate system S. The transformation rules for the electromagnetic field are well-known (β=v/c, γ=sqr(1−β˛) and c=1 assumed)):

(6)                                 E_x' = ¹/_γ(E_x − β B_y),         E_y' = ¹/_γ(E_y + β B_x),         E_z' = E_z ,
(7)                                 B_x' = ¹/_γ(B_x + β E_y),         B_y' = ¹/_γ(B_y − β E_x),         B_z' = B_z .

We shall check the first rule of Evans' O(3) symmetry Law (1.44) in our equivalent real formulation (5). Therefore we are now going to transform the wave (1-2) into the coordinate frame S' by means of the transformation rules (6-7) to obtain

(8)       B_x' = ^1−β/_γ B⁽⁰⁾ sqr(2) cos Φ = ^1−β/_γ B_x ,       B_y' = ^1−β/_γ B⁽⁰⁾ sqr(2) sin Φ = ^1−β/_γ B_y .       B_z' = B_z ,

which shows that due to

                ½ (B_x'˛ + B_y'˛) = ^1−β/_1+β ½ (B_x˛ + B_y˛) = ^1−β/_1+β B_z˛ = ^1−β/_1+β B_z'˛ < B_z'˛         (0 < β < 1)

Evans' first O(3)-symmetry relation (5) in S', the equation ½ (B_x'˛ + B_y'˛) = B_z'˛, is not fulfilled:

Evans' cyclical O(3)-symmetry is not Lorentz invariant and hence no law of Physics.