# Inconsistency in index contraction

Given an expression

$$g_{\mu\nu}v^{\nu} v^{\rho} k_{\nu} \delta^{\mu}_{\alpha} k_{\rho} \delta^{\nu}_{\beta}$$

if we contract $$v^{\rho} k_{\rho}$$, we get $$v \cdot k$$, then the rest of it reads $$g_{\mu\nu}v^{\nu} k_{\nu} \delta^{\mu}_{\alpha} \delta^{\nu}_{\beta}$$.

At this point, we can further contract $$v^{\nu} k_{\nu}$$ to get another factor of $$v \cdot k$$, and the rest contracts to $$g_{\alpha \beta}$$, so that the whole expression contracts to $$(v \cdot k)^2 g_{\alpha \beta}$$.

However, it seems that we can also contract $$g_{\mu\nu}v^{\nu} \delta^{\mu}_{\alpha}$$ and $$k_{\nu} \delta^{\nu}_{\beta}$$ separately to get $$(v \cdot k) v_{\alpha} k_{\beta}$$ as the final result.

In general, $$(v \cdot k) v_{\alpha} k_{\beta}$$ and $$(v \cdot k)^2 g_{\alpha \beta}$$ have different tensor structures. Where exactly does my derivation go wrong? I feel I might miss something obvious, but a hint is most appreciated.

The expression given makes no sense, since in Einstein notation each index can appear at most twice. In your expression the index $$\nu$$ shows up four times, and therefore one runs into ambiuities of what to contract with what. Where did this object come from? If it came from an earlier computation of yours, do it again taking care to never use the same name for two pairs of dummy indices.