Transmission failsafe i have a 1998 740il bmw transmission keeps going to failsafeand throws acode 106 speed sensor output what is my first step in fixin

Have you any idea of how many models and variant BMW have made in the year 2002?

Google

2002 BMW (model) output speed sensor location

and you will get this, but tailored to your particular mystery model.

https://www.google.com/search?q=%E2%80%8B2002+BMW+(model)+output+speed+sensor+location

..

the output speed sensor is located on the top of the drive extention housing. See:
https://www.realoem.com/bmw/enUS/showparts?id=DD63-USA-04-1998-E39-BMW-528i&diagId=24_1882
item 4

First check transmission dip stick for correct fluid level. Next schedule a transmission filter change. This should have been done at 60,000miles

The reason is the engine speed and transmission speed sensors don't agree. It could be caused by a bad transmission speed sensor, bad transmission fluid, a failed solenoid in the valve body ( in the transmission pan), a worn out valve body, a bad transmission, improper tires(that don't match each other), a faulty MAP sensor (air flow sensor on the air intake), faulty engine speed sensor and possibly a few other things.

Get the codes with a reader, and it might direct you in the right direction. But take the reading with a grain of salt, because it rarely tells you the exact problem. My fail safe came on and the code was TCC solenoid, but after replacing that, I had to install a rebuilt valve body to solve the problem.

code po720 refers to output shaft speed sensor (OSS) --circuit malfunction ===causes---wiring---OSS sensor---TCM
code po771 refers to shift solenoid (SS0 E --performance stuck off===causes --wiring ---shift solenoid.

speed sensor fault will throw those codes

The amber colored engine lamp means that something is wrong with either the engine or transmission systems. Input sensors monitor the engine and transmission to be sure they are within factory specifications. With this information the computer can adjust things like fuel control, idle speeds, transmission shift points etc. If an input or output is not working correctly the computer will sense this an throw an engine lamp on the inform you that there is a concern. In most cases the computer will use information from other sensors to keep the vehicle running until repairs are made. In more serious cases, the computer will flash the engine lamp and/or put the vehicle in fail safe mode. Fail safe mode will limit the amount of fuel to the fuel injectors and sometimes limit the transmission speeds and throttle response. You need to have codes retrieved either from a dealer, service station or auto parts store. The auto parts store will usually do it for free but their tester may not pull all codes. Good luck.

The transmission is going into failsafe mode, meaning the computer is seeing an electrical problem and turning off all solenoids locking it into on gear. Best bet is to get a scan tool and read the codes in the transmission module, common problems are the input and output speed sensors. If you are having problems with the speedometer, you probably need an output speed sensor. I've also had problems with pressure switch codes that have lead to replacing the valve body. If you have a speed sensor code i recommend replacing both the input and output speed sensors, they screw into the front side of the transmission.

I have 2002 elantra with similar problem. (but only takes 3 minutes to activate the problem) gets locked in 3rd gear, so have to start from a stop in 3rd, and revs really high when attempting freeway speed. also resets itself when you turn the ignition off, but will come back immediately (4 seconds, i think) as soon as you hit 30km/h the first time. (if you're seeing something slightly different read on anyway, some useful resources here)

So I'll start by saying I haven't fixed my problem yet, but found a lot of detail on it, which I'll share. First, check out http://www.hmaservice.com/ this is hyundai's site for service bulletins and other useful info. Locking into 3rd gear is one of 11 failsafe modes for the transmission, which they refer to as "electrical limp home mode" read the descriptions of the failsafe modes if yours is slightly different.
Second, get someone with an electronic read-out to check your DTCs (Diagnostic Trouble codes) I got mine done at AutoZone, who offers the service for free. Make sure you leave the car running, so the DTC doesn't clear itself, as it does each time you turn off the ignition. You'll probably get one of the four codes P0715, P0720, P0721 or P0722. 715 and 721 indicate a problem with the input speed sensor, which measures the rotation rate of the shaft going into the transmission, and the other two are codes for the output sensor, which reports the rate of rotation of the output shaft. In both cases, the problem can be that the sensor is blown or that the wiring leading up to it has a problem.

Now go back to hmaservice.com and look up TSB (technical service bulletin) 09-AT-016
it has a good picture of where to find the sensors on top of your transmission. there are some others that detail the failsafe modes (search on failsafe). you should have something to go on from here.

Unless your brother moved the sensor over from your old transmission to the new one, the fact that you still have the problem would suggest that the problem is in the wiring harness. Now, if it's a wiring problem, what would cause it to work OK for some time when the vehicle is cold? check to see if you have a puddle of coolant on the sensor when you start, and then look again after the problem kicks in. they put the sensors on top, where liquid can puddle, and beneath the coolant hoses where they often leak, especially after 8-9 years of use.

there is one catch, and that is if I read the info correctly, the 2000 elantra is a whole different series than the 2001-2009, and may be entirely different in codes, failsafes and sensor locations, but the three steps are still the best course for you.