cross-staff stems in a piano part

[Chris Jones]

I am attaching a .png file of the "before vs. after" type.

The top version is what I achieved with the following "simple example":

|   \version "2.18.2"
|   
|   global = {
|     \key g \major
|     \time 4/4
|   }
|   
|   right = \relative c'' {
|     \global
|     <<g4 d b>> <<g' d b>> <<fis' d>> <<fis c>>              |
|     % etc.
|   }
|   
|   left = \relative c' {
|     \global
|     \override Stem.direction = #DOWN
|     <<g4 g,>> <<g' g,>> <<a' d, d,>> <<a'' dis, dis,>>      |
|     % etc.
|   }
|   
|   \score {
|     \new PianoStaff \with {
|       instrumentName = "Piano"
|     } <<
|       \new Staff = "right" \with {
|       } \right
|       \new Staff = "left" \with {
|       } { \clef bass \left }
|     >>
|     \layout { }
|   }

The bottom version is the layout I need to create.

The difference between the two concerns the third beat of bar #1 where an
extra cross-staff stem is added between the D and the A below it.

Likewise regarding beat #4 where I want the C "connected" to the the A.

Naturally the stems connecting the two lower notes should no longer
extend to the the notes above!

How can I tell lilypond to produce this kind of layout?

All apologies for the obscure wording.

Thanks,

CJ