I'd run 2x10 outer joists with 2x6 rails inside, flush on the bottom. Suspend them from the 4x4 legs using [double concealed-flange hangers][1] set flush to the inside of the posts. This will leave an attractive 1/2" reveal at the outside.

The 2x6 rails will provide a ledge for your deck. Run 3 more 2x6 joists across the gap. 

I don't think the diagonals are necessary with this method. The hangers and tall joists will provide substantial diagonal bracing themselves, and your structure isn't that tall. Wind probably isn't much of a concern. 


  [1]: http://www.fastenersplus.com/Simpson-HUC210-2-Double-2x10-Concealed-Flange-Heavy-Face-Mount-Hanger-G90-Galvanized?gclid=Cj0KCQjw3MPNBRDjARIsAOYU6x8ZMiK2pqBSxYUdoxRTEP8eoj6bHAD7f_DMlmsyX-ipH9ou0tQmhRgaAlsSEALw_wcB