A possible proof for physicists of the theorem 5.2 is as follows:

We will use the following basic facts:

1. $ X = X _H + X_V$

2. $ \omega \left( X_{{H}} \right) =0$

3. $\omega \left( X_{{V}} \right) =X_{V}^* = constant$

4. $ Z \omega \left( X_{{V}} \right) =0 $

5. $ [X_H , Y_V] = Z_H$

5.a. $ \omega ([X_H , Y_V]) = \omega(Z_H) = 0$

6. $ \omega ([X_V , Y_V]) = [\omega(X_V) , \omega(Y_V) ] $

Proof : Given that $[X_V , Y_V]^* = [X_V ^*,Y_V ^*]$ then according with the fact 3 : $ \omega ([X_V , Y_V]) = [X_V , Y_V]^* $ and then : $ \omega ([X_V , Y_V]) = [X_V ^* , Y_V^* ] $; finally using again the fact 3 we obtain : $ \omega ([X_V , Y_V]) = [\omega(X_V) , \omega(Y_V) ] $.

7. $2 d\omega(X,Y)= X \omega(Y) - Y \omega(X) - \omega ([X,Y])$

8. $\Omega \left( X,Y \right) = d\omega(X_H,Y_H) $

From the fact 8 we write:

$$\Omega \left( X,Y \right) = d\omega(X_H,Y_H) $$

Using fact 1 we have: that

$$\Omega \left( X,Y \right) = d\omega( X - X_V,Y - Y_V) $$

which by bi-linearity is rewritten as

$$\Omega \left( X,Y \right) = d\omega(X , Y) - d\omega(X , Y_V) - d\omega(X_V , Y) + d\omega(X_V , Y_V) $$

Applying the fact 7 respectively to the three last terms of the right hand side of the last equation we have that

$$\Omega \left( X,Y \right) = d\omega(X , Y) - {\frac {1}{2}} X \omega(Y_V) + {\frac {1}{2}} Y_V \omega(X) + {\frac {1}{2}}\omega ([X,Y_V]) - $$

$${\frac {1}{2}} X_V \omega(Y) + {\frac {1}{2}} Y \omega(X_V) + {\frac {1}{2}}\omega ([X_V,Y]) + $$

$${\frac {1}{2}} X_V \omega(Y_V) - {\frac {1}{2}} Y_V \omega(X_V) - {\frac {1}{2}}\omega ([X_V,Y_V]) $$

Applying the fact 4 we have that : $X \omega(Y_V) = Y \omega(X_V) = X_V \omega(Y_V) =Y_V \omega(X_V) = 0 $. Using such results the main equation is reduced to

$$\Omega \left( X,Y \right) = d\omega(X , Y) + {\frac {1}{2}} Y_V \omega(X) + {\frac {1}{2}}\omega ([X,Y_V]) - $$

$${\frac {1}{2}} X_V \omega(Y) + {\frac {1}{2}}\omega ([X_V,Y]) - {\frac {1}{2}}\omega ([X_V,Y_V]) $$

Now, using the fact 1 in the second, third, fourth and fifth terms of the right hand side of the last equation we obtain that

$$\Omega \left( X,Y \right) = d\omega(X , Y) + {\frac {1}{2}} Y_V \omega(X_H + X_V) + {\frac {1}{2}}\omega ([X_H + X_V,Y_V]) - $$

$${\frac {1}{2}} X_V \omega(Y_H + Y_V) + {\frac {1}{2}}\omega ([X_V,Y_H+Y_V]) - {\frac {1}{2}}\omega ([X_V,Y_V]) $$

By linearity we have that

$$\Omega \left( X,Y \right) = d\omega(X , Y) + {\frac {1}{2}} Y_V \omega(X_H) + {\frac {1}{2}} Y_V \omega(X_V) + {\frac {1}{2}}\omega ([X_H ,Y_V]+[ X_V,Y_V]) - $$

$${\frac {1}{2}} X_V \omega(Y_H) - {\frac {1}{2}} Y_V \omega(Y_V) + {\frac {1}{2}}\omega ([X_V,Y_H]+[X_V,Y_V]) - {\frac {1}{2}}\omega ([X_V,Y_V]) $$

Using the facts 2 and 4 the last equation is reduced to

$$\Omega \left( X,Y \right) = d\omega(X , Y) + {\frac {1}{2}}\omega ([X_H ,Y_V]+[ X_V,Y_V]) + $$

$${\frac {1}{2}}\omega ([X_V,Y_H]+[X_V,Y_V]) - {\frac {1}{2}}\omega ([X_V,Y_V]) $$

Using again linearity we obtain that

$$\Omega \left( X,Y \right) = d\omega(X , Y) + {\frac {1}{2}}\omega ([X_H ,Y_V])+ {\frac {1}{2}}\omega([ X_V,Y_V]) + $$

$${\frac {1}{2}}\omega ([X_V,Y_H])+ {\frac {1}{2}}\omega([X_V,Y_V]) - {\frac {1}{2}}\omega ([X_V,Y_V]) $$

Simplifying the last equation we have that

$$\Omega \left( X,Y \right) = d\omega(X , Y) + {\frac {1}{2}}\omega ([X_H ,Y_V])+ {\frac {1}{2}}\omega([ X_V,Y_V]) + {\frac {1}{2}}\omega ([X_V,Y_H])$$

Using the fact 5 we obtain that

$$\Omega \left( X,Y \right) = d\omega(X , Y) + {\frac {1}{2}}\omega (Z_H)+ {\frac {1}{2}}\omega([ X_V,Y_V]) + {\frac {1}{2}}\omega (W_H)$$

Using again the fact 2, the last equation is reduced to

$$\Omega \left( X,Y \right) = d\omega(X , Y) + {\frac {1}{2}}\omega([ X_V,Y_V]) $$

Now using the fact 6 the last equation is transformed to

$$\Omega \left( X,Y \right) = d\omega(X , Y) + {\frac {1}{2}}[\omega(X_V) , \omega(Y_V) ] $$

Using the fact 1 in the second term of the right hand side of the last equation we have that

$$\Omega \left( X,Y \right) = d\omega(X , Y) + {\frac {1}{2}}[\omega(X - X_H) , \omega(Y - Y_H) ] $$

By linearity we obtain that

$$\Omega \left( X,Y \right) = d\omega(X , Y) + {\frac {1}{2}}[\omega(X) - \omega(X_H) , \omega(Y) -\omega (Y_H) ] $$

Finally using the fact 2 we derive that

$$\Omega \left( X,Y \right) = d\omega(X , Y) + {\frac {1}{2}}[\omega(X) , \omega(Y) ] $$

and the Theorem 5.2 is proved.

The structure equation (often called "the structure equation of Elie Cartan") is sometimes written, for the sake of simplicity, as follows:

$$\Omega = d\omega + {\frac {1}{2}}[\omega, \omega] $$

The corresponding expression for the Yang-Mills field is

$$F \left( X,Y \right) = dA(X , Y) + {\frac {1}{2}}[A(X) , A(Y) ] $$

or, for the sake of simplicity, as follows:

$$F = dA + {\frac {1}{2}}[A, A] $$