Problem 5:

3 - D rotations :

m0=DiagonalMatrix[{1,1,1}];
m1={{0,0,0},{0,0,-1},{0,1,0}};
m2={{0,0,1},{0,0,0},{-1,0,0}};
m3={{0,-1,0},{1,0,0},{0,0,0}};
MatrixForm /@ {m0,m1,m2,m3}

series= Series[Exp[-I θ m] -1 ,{θ,0,2}] //Normal

tmp1=series/.{(m^2)->m1.m1, m->m1};
tmp1//MatrixForm

tmp2=tmp1 + m0;
tmp2 //MatrixForm

rotx = tmp2 /.{1+((θ^2)/2)->Cos[θ],I θ->Sin[θ],-I θ->-Sin[θ]};
rotx //MatrixForm

3 - D rotations : commutator

comm[x_,y_]:=x.y-y.x

comm[m1,m2]- m3

comm[m2,m3]- m1

comm[m3,m1]- m2

3 - D rotations : Rotate about x-axis:

vec={x,y,z};
rotx.vec

Series[rotx.vec,{θ,0,1}] //Normal

3 - D Rotations: 2x2

s0={{1,0},{0,1}};
s1={{0,1},{1,0}};
s2={{0,-I},{I,0}};
s3={{1,0},{0,-1}};
MatrixForm /@ {s0,s1,s2,s3}

2 - D rotations : generator

tmp1= Series[ Exp[ + I θ/2 s] -1 ,{θ,0,2}] //Normal

tmp2=tmp1 /.{(s^2)-> s1.s1,s->s1};
tmp3=s0+tmp2;
tmp3 //MatrixForm

rules={(θ^2)->8(1-Cos[(θ/2)]),θ->2Sin[(θ/2)]};

srotx= tmp3 /.rules;
srotx //MatrixForm

2 - D rotations : commutator

(1/2)comm[s1,s2]- I s3

(1/2)comm[s2,s3]- I s1

(1/2)comm[s3,s1]- I s1

3 - D rotations : Rotate about x-axis:

vec2=x s1 + y s2 + z s3;
vec2 //MatrixForm

srotx //MatrixForm

Clear[conjugate]
conjugate[mat_]:=mat /.Complex[a_,b_]:> Complex[a,-b]

srotxCT=srotx //conjugate//Transpose;
srotxCT //MatrixForm

output =srotx.vec2.srotxCT //Simplify;
output //MatrixForm

vec2 //MatrixForm

Compare

vec={x,y,z};
rotx.vec