Show $[gamma^mu,eta^{nulambda}I]=0$ for Dirac matrices

I am trying to show the following commutation relation for the Dirac matrices $gamma^mu$ and the metric $eta^{nulambda}$: $$2[gamma^mu,eta^{nulambda}I]=0$$
where $I$ is the 4×4 identity matrix and $eta^{nulambda}$ is a matrix element of the metric.

The Dirac matrices satisfy the following anti-commutation relation:
$${gamma^mu,gamma^nu}= gamma^mu gamma^nu +gamma^nugamma^mu=2eta^{munu}I$$

Here is my attempt:
$$2[gamma^mu,eta^{nulambda}I]=2[gamma^mu,frac{1}{2}gamma^nu gamma^lambda +frac{1}{2}gamma^lambdagamma^nu]=[gamma^mu, gamma^nu gamma^lambda]+[gamma^mu,gamma^lambdagamma^nu]$$$$={gamma^mu, gamma^nu }gamma^lambda-gamma^nu{gamma^mu,gamma^lambda}+{gamma^mu,gamma^lambda }gamma^nu-gamma^lambda{gamma^mu,gamma^nu}$$$$=gamma^mugamma^nugamma^lambda-gamma^nugamma^lambdagamma^mu+gamma^mugamma^lambdagamma^nu-gamma^lambdagamma^nugamma^mu$$
where I cancelled like terms in the last step.

How do I proceed further to show that $2[gamma^mu,eta^{nulambda}I]=0$?