Raising and lowering indices

In Penrose abstract index notation, the expression "raising and lowering indices" refers to the following.
Suppose we have a Riemannian metric $g^{a b}$ . We want that, given $α_{b}$ , we can simply write $α^{a}$ to denote $g^{a b} α_{b}$ and, reciprocally, given $α^{b}$ , $α_{a}$ to denote $g_{a b} α^{b}$ . But we can have problems with tensors with both kind of indices (upper and lower). For example, to denote

g^{m z} α_{x y z}^{a b c}

what will we choose from $α_{x y}^{a b c m}$ , $α_{x y}^{a b m c}$ , $α_{x y}^{a m b c}$ , $α_{x y}^{m a b c}$ ?

We could take as a convention that we will always put the new index the first one ( $α_{x y}^{m a b c}$ ).

But this is not consistent because if we want to lower again we don't recover the original tensor. For example:

α_{x y w}^{a b c} = δ_{w}^{z} α_{x y z}^{a b c} = g_{w m} g^{m z} α_{x y z}^{a b c} = g_{w m} α_{x y}^{m a b c} = α_{w x y}^{a b c}

and this is not true in general.

For this reason, from now on we will denote tensors with some slots to indicate the order:

β_{x y}^{a}

Also, it can be shown easily that $δ_{b}^{a} = g_{b}^{a}$ and $δ_{a b} = g_{a b}$ .