In the Ricardian model, why are two out of 3 cases invalid?
To "specialize completely" means a country only produces one of the two goods.

Given your firms' (there seem to be four, one for each good in each country) profit maximization problems their optimality conditions are
$$
p_i \cdot \alpha_i^j \leq w^j
$$
with equality if $l_i^j &gt; 0$. From this, you can easily derive what parameter combinations of $p_i$ and $\alpha_i^j$ will result in conditions when a country specializes completely. 

I am not sure what you mean by 'fail', and the indices in your workforce notation seem a little off. Perhaps my detailed discussion of the Ricardian model can be of benefit. Beware it is not the exact same model as yours. For example, prices are not given exogenously, but they result from a supply and demand equilibrium.