r - Data frame to SpatialPolygonsDataFrame with multiple polygons

I have a data frame with a bunch of polygons, several points per polygon. How can I convert it to a SpatialPolygonsDataFrame object?

Here's my data frame:

> buildings_df
        lon      lat               id
1  96.13597 16.83363 convocation_hall

2  96.13580 16.83315 convocation_hall
3  96.13631 16.83298 convocation_hall
4  96.13656 16.83351 convocation_hall
5  96.13400 16.83104    judson_church
6  96.13392 16.83073    judson_church
7  96.13432 16.83064    judson_church
8  96.13444 16.83085    judson_church
9  96.13596 16.83074      the_library
10 96.13586 16.83044      the_library
11 96.13615 16.83035      the_library

12 96.13626 16.83065      the_library
13 96.13599 16.83073      the_library

This can be reproduced with:

> dput(buildings_df)
structure(list(lon = c(96.135972, 96.1358, 96.136315, 96.136562, 
96.133997, 96.133922, 96.134319, 96.134437, 96.135961, 96.135864, 
96.136154, 96.136261, 96.135993), lat = c(16.833625, 16.833153, 
16.832978, 16.833512, 16.831038, 16.830729, 16.830637, 16.830853, 
16.83074, 16.830442, 16.83035, 16.830647, 16.830729), id = structure(c(1L, 

1L, 1L, 1L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 3L), .Label = c("convocation_hall", 
"judson_church", "the_library"), class = "factor")), .Names = c("lon", 
"lat", "id"), row.names = c(NA, -13L), class = "data.frame")

From this data.frame, I then convert it to a list, then to polygons, then to a SpatialPolygons object

# make a list
buildings_list <- split(buildings_df, buildings_df$id)
# only want lon-lats in the list, not the names
buildings_list <- lapply(buildings_list, function(x) { x["id"] <- NULL; x })


#  make data.frame into spatial polygon, cf. http://jwhollister.com/iale_open_science/2015/07/05/03-Spatial-Data-In-R/

# create SpatialPolygons Object, convert coords to polygon
ps <- sapply(buildings_list, Polygon)

# add id variable 
p1 <- Polygons(ps, ID = 1) 

# create SpatialPolygons object
my_spatial_polys <- SpatialPolygons(list(p1), proj4string = CRS("+proj=longlat +datum=WGS84") ) 


# let's see them
plot(my_spatial_polys)

So far so good:

And each of the polygons seems to be in there...

> str(my_spatial_polys)
Formal class 'SpatialPolygons' [package "sp"] with 4 slots
  ..@ polygons   :List of 1

  .. ..$ :Formal class 'Polygons' [package "sp"] with 5 slots
  .. .. .. ..@ Polygons :List of 3
  .. .. .. .. ..$ convocation_hall:Formal class 'Polygon' [package "sp"] with 5 slots
  .. .. .. .. .. .. ..@ labpt  : num [1:2] 96.1 16.8
  .. .. .. .. .. .. ..@ area   : num 3.08e-07
  .. .. .. .. .. .. ..@ hole   : logi FALSE
  .. .. .. .. .. .. ..@ ringDir: int 1
  .. .. .. .. .. .. ..@ coords : num [1:5, 1:2] 96.1 96.1 96.1 96.1 96.1 ...
  .. .. .. .. ..$ judson_church   :Formal class 'Polygon' [package "sp"] with 5 slots
  .. .. .. .. .. .. ..@ labpt  : num [1:2] 96.1 43.3

  .. .. .. .. .. .. ..@ area   : num 0.0204
  .. .. .. .. .. .. ..@ hole   : logi TRUE
  .. .. .. .. .. .. ..@ ringDir: int -1
  .. .. .. .. .. .. ..@ coords : num [1:5, 1:2] 96.1 96.1 96.1 96.1 96.1 ...
  .. .. .. .. ..$ the_library     :Formal class 'Polygon' [package "sp"] with 5 slots
  .. .. .. .. .. .. ..@ labpt  : num [1:2] 96.1 43.3
  .. .. .. .. .. .. ..@ area   : num 0.00512
  .. .. .. .. .. .. ..@ hole   : logi TRUE
  .. .. .. .. .. .. ..@ ringDir: int -1
  .. .. .. .. .. .. ..@ coords : num [1:6, 1:2] 96.1 96.1 96.1 96.1 96.1 ...

  .. .. .. ..@ plotOrder: int [1:3] 1 2 3
  .. .. .. ..@ labpt    : num [1:2] 96.1 16.8
  .. .. .. ..@ ID       : chr "1"
  .. .. .. ..@ area     : num 0.0302
  ..@ plotOrder  : int 1
  ..@ bbox       : num [1:2, 1:2] 96.1 16.8 96.1 16.8
  .. ..- attr(*, "dimnames")=List of 2
  .. .. ..$ : chr [1:2] "x" "y"
  .. .. ..$ : chr [1:2] "min" "max"
  ..@ proj4string:Formal class 'CRS' [package "sp"] with 1 slot

  .. .. ..@ projargs: chr "+proj=longlat +datum=WGS84 +ellps=WGS84 +towgs84=0,0,0"

But they don't seem to be grouped in the way I'm expecting. When I try to set three ID variables, I get an error. How can I get the SpatialPolygons to have three polygons, each with its own ID?