Why are allocation and distribution important consequences of the second welfare theorem? I'm reading Intermediate Microeconomics: A Modern Approach by Hal Varian. 
On page 606, he states:


  The Second Theorem of Welfare Economics states that as long as preferences are convex, then every Pareto efficient allocation can be supported as a competitive equilibrium. 


He then states on 604,


  The Second Welfare Theorem implies that the problems of distribution and efficiency can be separated...Prices play two roles in the market system system: an allocative role and a distributive role. The allocative role of prices is to indicate relative scarcity; the distributive role is to determine how much of different goods different agents can purchase. The Second Welfare Theorem says that these two roles can be separated: we can redistribute endowments of goods to determine how much wealth agents have, and then use prices to indicate relative scarcity. 


My Question:

How does the second remark follow from the first?
The First Welfare Theorem essentially says


  price equilibrium with transfers => Pareto optimal


The FWT hence solves the "problem of efficiency" i.e. If we let the market work through prices, we will achieve Pareto efficiency.

The SWT, on the other hand, states


  Pareto optimal => price quasi-equilibrium with transfers* 


It assures us that any Pareto optimal point can be achieved via the market. 

Note that distribution and inequality are normative problems which must be addressed through say, the political process, rather than economics. The concept of Pareto efficiency does not tell us anything about distribution. Oftentimes in a 2x2 model, giving Agent 1 all the resources and Agent 2 none of the resources counts as Pareto optimal.

Therefore the SWT addresses the "problem of distribution" by saying that "Ok, society has decided that a certain Pareto optimum is the most desirable outcome. We know that we can achieve this by redistributing endowments/incomes and then letting the market work."

*PQWT is a weaker definition of PEWT