3

enter image description here

\language "english"

global = {
  \key c \minor
  \numericTimeSignature \time 4/4
}

right = \relative c'' {
  \global
  < g c d g >1\arpeggio \bar "||"
}

left = \relative c, {
  \global
  << \crossStaff {r4 d'8 ( g8 \lt; c8 d8 g8 c,8 d2.) <g, g'>4 \f } \\ {g,1 g4 ( d'4 g4 g,4)} >> \bar "||"
}

\score {
  \new PianoStaff <<
    \new Staff = "right" \right
    \new Staff = "left" { \clef bass \left }
  >>
  \layout {
      \context {
      \PianoStaff
      \consists #Span_stem_engraver
    }
  }
}

enter image description here

2

Nice starting code: that alone deserves an answer. Try the following:

\language "english"

global = {
  \key c \minor
  \numericTimeSignature \time 4/4
}

right = \relative c'' {
  \global
  < g c d g >1\arpeggio \bar "||"
}

left = \relative c, {
  \global
  << {r4 d'8 ( g8
      \change Staff = "right" \voiceTwo c8\< d8 g8 c,8 \oneVoice b2.) <g' g'>4 \f } \\
     {g,,1 \oneVoice g4 ( d'4 g4 g,4)}
  >> \bar "||"
}

\score {
  \new PianoStaff <<
    \new Staff = "right" \right
    \new Staff = "left" { \clef bass \left }
  >>
  \layout {
      \context {
      \PianoStaff
      \consists #Span_stem_engraver
    }
  }
}

The decisive thing is the \change Staff directive here. Also there are a few directives in order to use the \voiceOne/\voiceTwo distinction implied by <<...\\...>> only where there are actually two voices in one Staff.

The result does not use the nicest slur curvature but is reasonably fine: image of music

This does not actually use the \crossStaff command intended for stems or the respective engraver.

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