Let $E$ be a complex vector bundle of rank $r$ and suppose we can write $E = \oplus_{i=1}^r L_i$ where $L_i$ are line bundles. I have read here (and think I more or less understand why) that the total chern class of $E$ can be written in this case as:

$$c(E) = \prod_{i=1}^r(1+c_1(L_i))$$

My question: is there any similar simple expression for $c_1(E)$? In particular, is it true that in this case I can write $c_1(E) = \sum_{i=1}^r c_1(L_i)$? Thanks in advance!