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Davis and Jedwab (1997) established a great construction theory unifying manypreviously known constructions of difference sets, relative difference sets anddivisible difference sets. They introduced the concept of building blocks,which played an important role in the theory. On the other hand, Polhill (2010)gave a construction of Paley type partial difference sets (conference graphs)based on a special system of building blocks, called a covering extendedbuilding set, and proved that there exists a Paley type partial difference setin an abelian group of order $9^iv^4$ for any odd positive integer $v>1$ andany $i=0,1$. His result covers all orders of nonelementary abelian groups inwhich Paley type partial difference sets exist. In this paper, we give newconstructions of strongly regular Cayley graphs on abelian groups by extendingthe theory of building blocks. The constructions are large generalizations ofPolhill's construction. In particular, we show that for a positive integer $m$and elementary abelian groups $G_i$, $i=1,2,\ldots,s$, of order $q_i^4$ suchthat $2m\,|\,q_i+1$, there exists a decomposition of the complete graph on theabelian group $G=G_1\times G_2\times \cdots\times G_s$ by strongly regularCayley graphs with negative Latin square type parameters $(u^2,c(u+1),- u+c^2+3c,c^2+ c)$, where $u=q_1^2q_2^2\cdots q_s^2$ and $c=(u-1)/m$. Such stronglyregular decompositions were previously known only when $m=2$ or $G$ is a$p$-group. Moreover, we find one more new infinite family of decompositions ofthe complete graphs by Latin square type strongly regular Cayley graphs. Thus,we obtain many strongly regular graphs with new parameters.

New constructions of strongly regular Cayley graphs on abelian groups