Let $A=K[X_1, X_2,dots, X_8]$ be the polynomial algebra in 8 variables and $K$ is an algebraically closed field of characteristic zero. Let $a_1=X_3X_4, a_2=X_5X_6, a_3=X_7X_8$, and $a_4=X_7X_4X_6$.
I have two questions:

prove that the sum $Sigma_{n_1, dots, n_6geq 0}KX_1^{n_1}X_2^{n_2}a_1^{n_3}a_2^{n_4}a_3^{n_5}a_4^{n_6}$ is dirct sum if and only if the monomials $X_1, X_2, a_1, a_2, a_3, a_4$ are algebraically independent in the polynomial algebra $A$.

prove that the monomials $X_1, X_2, a_1, a_2, a_3, a_4$ are algebraically independent in the polynomial algebra $A$.
Thank you for any help