Suppose $E \to B $ is a principal $G$-bundle with connection $\omega \in \Omega^1(E,\mathfrak{g})$, and corresponding curvature $\Omega \in \Omega^2(E,\mathfrak{g})$. Then $\Omega^*(E)$ is a graded-differential algebra, with the exterior derivative as the differential, and wedge product of forms being the multiplication.

Let $\omega_i$'s (resp. $\Omega_i$'s) be the connection 1-forms (resp. curvature 2-forms) on $E$ w.r.t. the chosen basis of $\mathfrak{g}$. Thus $\omega=\sum_{i=1}^n \omega_i e_i $ and $\Omega=\sum_{i=1}^n \Omega_i e_i $

There is an algebra homomorphism $f: W(\mathfrak{g}^*) \to \Omega^*(E)$ given by $\theta_i \mapsto \omega_i$, and $ u_i \mapsto \Omega_i$ (for clarity of notation, I am using $u_i$'s as generators of $S(\mathfrak{g}^*)$ instead of your notation $\Omega_i$).

Defining the Weil-differential in the manner it is defined makes sure that this algebra homomorphism is a graded-differential algebra homomorphism (i.e. is compatible w.r.t. the derivation operation) by the virtue of the 'structure equation' : $\Omega=d\omega + \frac{1}{2}[\omega,\omega]$ and the 'Bianchi identity' $d\Omega=\omega \wedge \Omega$

Suppose $E \to B $ is a principal $G$-bundle with connection $\omega \in \Omega^1(E,\mathfrak{g})$, and corresponding curvature $\Omega \in \Omega^2(E,\mathfrak{g})$. Then $\Omega^*(E)$ is a differential graded algebra, with the exterior derivative as the differential, and wedge product of forms being the multiplication.

Let $\omega_i$'s (resp. $\Omega_i$'s) be the connection 1-forms (resp. curvature 2-forms) on $E$ w.r.t. the chosen basis of $\mathfrak{g}$. Thus $\omega=\sum_{i=1}^n \omega_i e_i $ and $\Omega=\sum_{i=1}^n \Omega_i e_i $

There is an algebra homomorphism $f: W(\mathfrak{g}^*) \to \Omega^*(E)$ given by $\theta_i \mapsto \omega_i$, and $ u_i \mapsto \Omega_i$ (for clarity of notation, I am using $u_i$'s as generators of $S(\mathfrak{g}^*)$ instead of your notation $\Omega_i$).

Defining the Weil-differential in the manner it is defined makes sure that this algebra homomorphism is a differential graded algebra homomorphism (i.e. is compatible w.r.t. the derivation operation) by the virtue of the 'structure equation' : $\Omega=d\omega + \frac{1}{2}[\omega,\omega]$ and the 'Bianchi identity' $d\Omega=\omega \wedge \Omega$

Suppose $E \to B $ is a principal $G$-bundle with connection $\omega \in \Omega^1(B,\mathfrak{g})$, and corresponding curvature $\Omega \in \Omega^2(B,\mathfrak{g})$. Then $\Omega^*(E)$ is a graded-differential algebra, with the exterior derivative as the differential, and wedge product of forms being the multiplication.

Let $\omega_i$'s (resp. $\Omega_i$'s) be the connection 1-forms (resp. curvature 2-forms) on $E$ w.r.t. the chosen basis of $\mathfrak{g}$. Thus $\omega=\sum_{i=1}^n \omega_i e_i $ and $\Omega=\sum_{i=1}^n \Omega_i e_i $

There is an algebra homomorphism $f: W(\mathfrak{g}^*) \to \Omega^*(E)$ given by $\theta_i \mapsto \omega_i$, and $ u_i \mapsto \Omega_i$ (for clarity of notation, I am using $u_i$'s as generators of $S(\mathfrak{g}^*)$ instead of your notation $\Omega_i$).

Defining the Weil-differential in the manner it is defined makes sure that this algebra homomorphism is a graded-differential algebra homomorphism (i.e. is compatible w.r.t. the derivation operation) by the virtue of the 'structure equation' : $\Omega=d\omega + \frac{1}{2}[\omega,\omega]$ and the 'Bianchi identity' $d\Omega=\omega \wedge \Omega$

Suppose $E \to B $ is a principal $G$-bundle with connection $\omega \in \Omega^1(E,\mathfrak{g})$, and corresponding curvature $\Omega \in \Omega^2(E,\mathfrak{g})$. Then $\Omega^*(E)$ is a graded-differential algebra, with the exterior derivative as the differential, and wedge product of forms being the multiplication.

Let $\omega_i$'s (resp. $\Omega_i$'s) be the connection 1-forms (resp. curvature 2-forms) on $E$ w.r.t. the chosen basis of $\mathfrak{g}$. Thus $\omega=\sum_{i=1}^n \omega_i e_i $ and $\Omega=\sum_{i=1}^n \Omega_i e_i $

There is an algebra homomorphism $f: W(\mathfrak{g}^*) \to \Omega^*(E)$ given by $\theta_i \mapsto \omega_i$, and $ u_i \mapsto \Omega_i$ (for clarity of notation, I am using $u_i$'s as generators of $S(\mathfrak{g}^*)$ instead of your notation $\Omega_i$).

Defining the Weil-differential in the manner it is defined makes sure that this algebra homomorphism is a graded-differential algebra homomorphism (i.e. is compatible w.r.t. the derivation operation) by the virtue of the 'structure equation' : $\Omega=d\omega + \frac{1}{2}[\omega,\omega]$ and the 'Bianchi identity' $d\Omega=\omega \wedge \Omega$