Infinitesimal Lorentz transformation

I have questions about the infinitesimal Lorentz transformation. but specifically about index manipulations.

$$\Lambda^{\mu}_{}_{\nu}=\delta^{\mu}_{\nu}+\delta\omega^{\mu}_{}_{\nu}$$
where $$\delta\omega^{\mu}_{}_{\nu} << 1$$

as found in many textbooks, we substitute this into

$$g_{\mu\nu}\Lambda^{\mu}_{}_{\alpha}\Lambda^{\nu}_{}_{\beta}=g_{\alpha\beta}$$

and do the tedious(?) algebra...

$$g_{\mu\nu}(\delta^{\mu}_{\alpha}\delta^{\nu}_{\beta}+\delta^{\mu}_{\alpha}\delta\omega^{\nu}_{}_{\beta}+\delta\omega^{\mu}_{}_{\alpha}\delta^{\nu}_{\beta}+\delta\omega^{\mu}_{}_{\alpha}\delta\omega^{\nu}_{}_{\beta})=g_{\alpha\beta}$$

and the last term is negligible because it too small (correct me if wrong!)

my question is the second and third terms. they are supposed to become...

$$...\delta\omega_{\mu\nu}+\delta\omega_{\nu\mu}...$$

why is that? because $$\delta^{\mu}_{\nu}'s$$ are identity matrices, so whichever order we multiply, we get the same result.
and I look up some articles and found something about "abstract index notation". but is this the one?

malawi_glenn
Homework Helper
$$g_{\mu\nu}\Lambda^{\mu}_{}_{\alpha}\Lambda^{\nu}_{ }_{\beta}=g_{\alpha\beta}$$

should be

$$g_{\nu\mu}\Lambda^{\mu}_{}_{\alpha}\Lambda _{\beta}{}^{\nu} =g_{\alpha\beta}$$

I think the one -at certain stages- shoul not worry about similar metters, since this consideration of infinitesimal rotation is usefal for a quick conviction that ''omega"" are asymetric.I do not kow how to proof it exactly but Idid believe my profesor when he said that we can generalize that for even noninfinitisimal ransformation.
I think her is no more than a mathemrical work needed, wich in dead, may be every where in the text books( of high math possibly).

thanks everyone.
malawi_glenn, your explanation made me realize the important point! although I have exactly copied that part from a textbook...

thanks again.

malawi_glenn
$$g = \Lambda^{T} g \Lambda$$