The Lower Brook rose as the drainage from several ponds situated close to Edge Lane and to the east of what is now Botanic Park but which earlier in its history was the Botanic gardens shown on this map.  These ponds existed as recently as 1902, perhaps later as ponds are still shown in the area in the 1960s (IF they are the same ponds that is). The original scan of is map, the 1849 City Engineer's map of Liverpool, was kindly sent to me by Richard Hawes of lancashire galleries.  I have coloured this (and rotated it to save space).  It suggests that there are two streams in the upper reaches of where the lower brook rose.

The historic course of the Lower Brook was severed by the Liverpool to Manchester railway lines at Edge Hill (shown left · · · · · · · · with the word 'LIVERPOOL' along it).  In the 1700s it continued and crossed Wavertree Road close to a house called 'Bridge House'. J. A. Picton, in 1875, records this as follows " ...1830.... Just where the railway crosses the road stood a farm-house and orchard called Bridge House, from the bridge over the brook forming the division between the townships. About half way from Edgehill stood an old picturesque cottage, originally called the 'Pump House' but subsequently converted to a tavern called the Halfway House, with a tea garden and skittles". Bridge house would have been found about (and below!) the middle of the present road bridge over the railway lines.

The stream appears to have followed the line of what is now Spofforth Road and Webster Road.  This line forms part of what was then a Parliamentary boundary - which I do not think is co-incidental - this happens frequently.  In addition both of these roads run at a noticeable angle to the surrounding terraces, which, in every case within Toxteth, is an indicator of the presence of an ancient stream 

As shown here (extreme left), the Lower Brook crossed Smithdown Road at what is now the entrance to Toxteth Park Cemetery, near Salisbury Road.  This disagrees with Griffiths in 1907 (and others who have used his excellent works as a source) but I think that he is wrong. Griffiths gives Mulliner Street as the crossing point on Smithdown Road  I find this totally implausible based on geography and contours.

Mulliner Street is at one of the steepest downhill gradients on Smithdown Road and I do not believe that the Lower Brook would cross Smithdown Road here. To do so the brook would run sideways and slightly up, a downward running hill. Griffiths suggested crossing and direction is marked blue on the [clickable] pictures.

If you visit the junction you will also notice that Mulliner Street slopes downhill away from Smithdown Road, so a crossing at Mulliner Street  would not only need to flow sideways up a hill but would have to flow uphill along Mulliner Street to do so!   A stream is more likely to have followed contours and been situated at the lowest point, which is is around the Cemetery gates.  I have now satisifed myself that this is the crossing point.  This was done by re-sizing the 1765, Earl of Sefton, map and overlaying it onto a recent OS map.  Lining up portions of the streams still above ground and surviving road junctions shows the Lower Brook / Smithdown Road crossing to be by the Cemetery gates, the Lower Brook would flow close to Webster or Cranborne Road (perhaps Salisbury or Alderson Road) before crossing Smithdown Road. This is the lowest point of Smithdown Road here - I checked. (This is also the point where the boundary of Toxteth · · · · · · · · switches from the Cemetery side of Smithdown Road to the opposite side, prior to diverting away from Smithdown Road for a short distance, rejoing at the present Brook House).

After passing under the cemetery, in a drain, it crosses Ullet Road somewhere near Cheltenham Avenue and enters what is now Sefton Park.  It finally emerges above ground at The Grotto. It can clearly been seen there as a small open stream as late as the 1905 map and of course the water flow is still there regardless of how it enters the grotto pool.  The watercourse is above ground now and flows through Sefton Park and into the head of the main boating lake. Here it merges with the waters of the Upper Brook, something that it always did at this point  and this can be seen on the map, left.  The combined waters leave at the far end of the lake and flow in a culvert under Aigburth Road and then flow more or less down the course of the old Osklesbrok through the very clearly shown 'Otterspool' and finally on into the Mersey.

(The right hand stream on the map opposite is the Upper Brook which has a file of its own, although you can pick out the crossing point at Smithdown Road (by the Brook House), the dammed pool which is now in Greenbank Park and the junction with the Lower Brook).

 
As a postscript to my disagreement with Griffiths' work, I have examined a variety of early maps, some of which show several interesting things.  There is clearly a farm or smallholding at about the junction of Mulliner Street and Smithdown Road.  There are also ponds on the Hartington Road side of Smithdown Road in this same area. The ponds and Hartington Road are higher than Mulliner Street and there is a suggestion on some that there could have been a small brook here. This is not  the Lower Brook, but would have crossed Smithdown Road in the reverse direction to the Lower Brook and thus it would flow down Mulliner Street, down the observed gradient. The course appears very straight and it may have been a channel or ditch which provided water for the smallholding.