Let $a,b,c,d \in R$ be four (probably different) elements of a commutative $k$-algebra $R$, where $k$ is a field of characteristic zero.
Obviously, $\langle a+b,c+d \rangle \subseteq \langle a,b,c,d \rangle$.
When $\langle a+b,c+d \rangle = \langle a,b,c,d \rangle$?
An example in a ring $R$ with involution $\beta$ (= an automorphism of $R$ of order two): If the following conditions are satisfied, then $\langle a+b,c+d \rangle = \langle a,b,c,d \rangle$:
(1) $a,c$ are symmetric.
(2) $b,d$ are skew-symmetric.
(3) $I:=\langle a+b,c+d \rangle$ is invariant under $\beta$.
Indeed, $\beta(I) \subseteq I$ (= $I$ is invariant under $\beta$) implies that $a-b=\beta(a+b) \in I$ and then $(a-b)+(a+b)=2a \in I$, so $a \in I $ and also $b=(a+b)-a \in I$. Similarly, $c,d \in I$.
Thank you very much!