Ordinary math mode enclosed by display math mode

Question

Purely academic TeX question! The solution of Exercise 13.5 in The TeXbook

Vertical mode can occur only as the outermost mode; horizontal mode and display math mode can occur only when immediately enclosed by vertical orinternal vertical mode; ordinary math mode  cannot be immediately enclosed by vertical or internal vertical mode; all other cases are possible.

seems to suggest that display math mode can directly enclose ordinary math mode... but how? The naiive approach with
$$Display math $ordinary math$ display math$$

won't work:
! Display math should end with $$.
<to be read again> 
                   o
l.1 $$Display math $o
                     rdinary math$ display math$$

David Carlisle · Accepted Answer

tracingonline2

$$aeqno{1+2showlists}$$

bye

shows
### math mode entered at line 6
mathord
.fam0 1
mathbin
.fam0 +
mathord
.fam0 2
### math mode entered at line 6
mathord
### display math mode entered at line 6
mathord
.fam1 a
### vertical mode entered at line 0

so Vertical mode containing Display math containing Math.
Actually as that output shows that is two nested math modes in display math. To get one level of math mode it is enough to do
tracingonline2

$$a{showlists}$$

bye

which logs
### math mode entered at line 5
### display math mode entered at line 5
mathord
.fam1 a
mathord
### vertical mode entered at line 0

or

tracingonline2

$$aeqno 1showlists$$

bye

which is
### math mode entered at line 5
mathord
.fam0 1
### display math mode entered at line 5
mathord
.fam1 a
### vertical mode entered at line 0

Ordinary math mode enclosed by display math mode

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