Sprawdzenie wzoru Rodriguesa 
http://mathworld.wolfram.com/RodriguesRotationFormula.html

n={n1,n2,n3};

minusiX1={{0,0,0},{0,0,-1},{0,1,0}};

minusiX2={{0,0,1},{0,0,0},{-1,0,0}};

minusiX3={{0,-1,0},{1,0,0},{0,0,0}};

MatrixForm[minusiX1]

MatrixForm[minusiX2]

MatrixForm[minusiX3]

minusiXM={minusiX1,minusiX2,minusiX3};

A=n.minusiXM

MatrixForm[A]

A2=FullSimplify[A.A,Element[{n1,n2,n3},Reals]&&(n1*n1+n2*n2+n3*n3==1)]

MatrixForm[A2]

A2={{-1+(n1^2),n1 n2,n1 n3},{n1 n2,-1+(n2^2),n2 n3},{n1 n3,n2 n3,-1+(n3^2)}}

MatrixForm[A2]

A3=FullSimplify[A2.A,Element[{n1,n2,n3},Reals]&&(n1*n1+n2*n2+n3*n3==1)]

MatrixForm[A3]

FullSimplify[A3==-A,Element[{n1,n2,n3},Reals]&&(n1*n1+n2*n2+n3*n3==1)]

A4=FullSimplify[A3.A,Element[{n1,n2,n3},Reals]&&(n1*n1+n2*n2+n3*n3==1)]

MatrixForm[A4]

FullSimplify[A3.A==-A2,Element[{n1,n2,n3},Reals]&&(n1*n1+n2*n2+n3*n3==1)]

A5=FullSimplify[-A2.A,Element[{n1,n2,n3},Reals]&&(n1*n1+n2*n2+n3*n3==1)]

MatrixForm[A5]

FullSimplify[A5==A,Element[{n1,n2,n3},Reals]&&(n1*n1+n2*n2+n3*n3==1)]

A6=FullSimplify[A5.A,Element[{n1,n2,n3},Reals]&&(n1*n1+n2*n2+n3*n3==1)]

MatrixForm[A6]

FullSimplify[A6==A2,Element[{n1,n2,n3},Reals]&&(n1*n1+n2*n2+n3*n3==1)]

A7=FullSimplify[A2.A,Element[{n1,n2,n3},Reals]&&(n1*n1+n2*n2+n3*n3==1)]

MatrixForm[A7]

FullSimplify[A7==-A,Element[{n1,n2,n3},Reals]&&(n1*n1+n2*n2+n3*n3==1)]

A8=FullSimplify[-A.A,Element[{n1,n2,n3},Reals]&&(n1*n1+n2*n2+n3*n3==1)]

MatrixForm[A8]

FullSimplify[A8==-A2,Element[{n1,n2,n3},Reals]&&(n1*n1+n2*n2+n3*n3==1)]

A9=A8.A;

FullSimplify[A9==A,Element[{n1,n2,n3},Reals]&&(n1*n1+n2*n2+n3*n3==1)]

A10=A9.A;

FullSimplify[A10==A2,Element[{n1,n2,n3},Reals]&&(n1*n1+n2*n2+n3*n3==1)]

(* A może tak się uda ? *)

R=IdentityMatrix[3] + Sin[phi]*A + (1-Cos[phi])*A2;

FullSimplify[MatrixExp[phi*A]==R,Element[{n1,n2,n3},Reals]&&(n1*n1+n2*n2+n3*n3==1)]

minusiX1={{0,0,0},{0,0,-1},{0,1,0}};

minusiX2={{0,0,1},{0,0,0},{-1,0,0}};

minusiX3={{0,-1,0},{1,0,0},{0,0,0}};

minusiX1.minusiX2-minusiX2.minusiX1==minusiX3

minusiX2.minusiX3-minusiX3.minusiX2==minusiX1

minusiX3.minusiX1-minusiX1.minusiX3==minusiX2

commutator[a_,b_]:=a.b-b.a

commutator[commutator[minusiX1,minusiX2],minusiX3]

commutator[commutator[minusiX2,minusiX3],minusiX1]

commutator[commutator[minusiX3,minusiX1],minusiX2]

X1=(-1/I)*minusiX1

X2=(-1/I)*minusiX2

X3=(-1/I)*minusiX3

commutator[X1,X2]==I*X3

commutator[X2,X3]==I*X1

commutator[X3,X1]==I*X2

commutator[commutator[X1,X2],X3]

commutator[commutator[X2,X3],X1]

commutator[commutator[X3,X1],X2]