after some calculations I obtain the next expression

$$\lambda^{c}\nabla_{a}g_{bc}=\lambda^{c}{}(\lambda_{c}\nabla_{a}\lambda_{b}-g_{be} \Gamma^{e}_{ad}\lambda^{d}\lambda_{c}) $$ So my question is if I can eliminate $\lambda^{c}$ of this equation, and get something like this

$$\nabla_{a}g_{bc}=\lambda_{c}\nabla_{a}\lambda_{b}-g_{be} \Gamma^{e}_{ad}\lambda^{d}\lambda_{c} $$