Given a symmetric, bounded, elliptic bilinear form $a(\cdot,\cdot)$ on a space $V=V_1 \oplus V_2$ I want to show that there is a constant $c$ such that $$ \forall v = v_1 + v_2 \in V_1 \oplus V_2: a(v_1,v_1) + a(v_2,v_2) \leq c \cdot a(v_1+v_2,v_1+v_2)$$ (with $v_1 \in V_1, v_2 \in V_2)$.
By the properties of $a$, we get an induced norm $||\cdot||=\sqrt{a(\cdot,\cdot)}$ on $V$. So, above condition translates to (sort of) an inverse of the triangle inequality $$ ||v_1||^2+||v_2||^2 \leq c \cdot ||v_1+v_2 ||^2. $$ All the things I tried did not help because the estimates were always in the wrong direction. A useful tool might be to use $$\alpha = \sup_{0 \neq v_1 \in V_1, 0\neq v_2 \in V_2} \tfrac{a(v_1,v_2)}{||v_1|| \cdot ||v_2||} \in [0,1)$$ but I 'm not sure how it could help.