textbook: 3.25
<<LorentzBoost.m  
b=.25
gamma=1/Sqrt[1-(b^2+b^2)]    
U0=gamma {b,b,0,I} 
Out[4]= {0.267261, 0.267261, 0., 0. + 1.06904 I}

U1=boost[b,0,0].U0
Out[5]= {0. + 0. I, 0.267261 + 0. I, 0. + 0. I, 0. + 1.0351 I}

V1y=U1[[2]]/U1[[4]] I
Out[6]= 0.258199 + 0. I

invariants_Fuv.pdf #2, #3

fuv={{0,b3,-b2,-I e1},{-b3,0,b1,-I e2},{b2,-b1,0,-I e3},{I e1,I e2,I e3,0}}    
fpuv=boost[v,0,0].fuv.Transpose[boost[v,0,0]]
MatrixForm[Simplify[%]]

Out[10]//MatrixForm= 
 
>                     b3 - e2 v        -b2 - e3 v
                     ------------     ------------
                               2                2
    0                Sqrt[1 - v ]     Sqrt[1 - v ]     -I e1

     -b3 + e2 v                                        I (-e2 + b3 v)
    ------------                                       --------------
              2                                                   2
    Sqrt[1 - v ]     0                b1                Sqrt[1 - v ]

     b2 + e3 v                                         -I (e3 + b2 v)
    ------------                                       --------------
              2                                                   2
    Sqrt[1 - v ]     -b1              0                 Sqrt[1 - v ]

                     I (e2 - b3 v)    I (e3 + b2 v)
                     -------------    -------------
                               2                2
    I e1             Sqrt[1 - v ]     Sqrt[1 - v ]     0


In[11]:= Simplify[Tr[fuv.fuv]]                                                  

               2     2     2     2     2     2
Out[11]= -2 (b1  + b2  + b3  - e1  - e2  - e3 )

In[12]:= Simplify[Tr[fpuv.fpuv]]                                                

               2     2     2     2     2     2
Out[12]= -2 (b1  + b2  + b3  - e1  - e2  - e3 )

In[13]:= Simplify[Det[fuv]]                                                     

                                 2
Out[13]= -(b1 e1 + b2 e2 + b3 e3)

In[14]:= Simplify[Det[fpuv]]                                                    

                                 2
Out[14]= -(b1 e1 + b2 e2 + b3 e3)

In[15]:= Simplify[Det[fuv.fuv]]                                                 

                                4
Out[15]= (b1 e1 + b2 e2 + b3 e3)

In[16]:= Simplify[Det[fpuv.fpuv]]                                               

                                4
Out[16]= (b1 e1 + b2 e2 + b3 e3)

Result: E^2-B^2 & E.B are invarient.
Note that E=B and E perpendicular to B are characteristics of light,
so transformed light will still look like light