
lebenslänglich
life
In a legal sense, it refers to life imprisonment, the most severe punishment in German criminal law.
Der Mörder wurde zu einer lebenslänglichen Haftstrafe verurteilt.
The murderer was sentenced to life imprisonment.
life / lifelong
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life
In a legal sense, it refers to life imprisonment, the most severe punishment in German criminal law.
Der Mörder wurde zu einer lebenslänglichen Haftstrafe verurteilt.
The murderer was sentenced to life imprisonment.

lifelong
Lasting or valid for an entire lifetime.
Sie erhielt ein lebenslängliches Wohnrecht in dem Haus.
She received a lifelong right of residence in the house.
The word is made up of 'Leben' (life), in the form 'lebens-', and the adjective stem 'lang' (long), plus the ending '-lich'. So it literally means 'for a lifetime'.
Er hat eine lebenslängliche Garantie auf das Produkt.
He has a lifelong guarantee on the product.
'Lebenslänglich' (for life/lifelong) is very often used like an adverb without a declension ending, especially in legal contexts. It then answers the question 'Wie lange?' (for how long?).
Der Angeklagte wurde zu lebenslänglich verurteilt.
The defendant was sentenced to life.
In everyday usage, 'lebenslänglich' (life/for life) almost always refers to life imprisonment. The noun 'Haft' (imprisonment) or 'Strafe' (sentence/punishment) is often omitted because the context is clear.
Er hat für den Mord lebenslänglich bekommen.
He got life for the murder.
Outside criminal law, 'lebenslänglich' (lifelong) is often combined with rights or obligations. Typical examples are a 'lebenslängliches Wohnrecht' (lifelong right of residence) or a 'lebenslängliche Rente' (lifelong pension).
Ihre Großmutter hat ein lebenslängliches Nießbrauchrecht für das Haus.
Her grandmother has a lifelong usufruct right to the house.
lifelong / life
Both words can refer to a prison sentence lasting until the end of the prisoner's life. Lebenslang is the more general form and can also describe a lifelong duration outside the legal context. Lebenslänglich is especially established as a term for the sentence imposed and is therefore used mainly when referring to a verdict.

Das Gericht verurteilte ihn zu lebenslänglich.
The court sentenced him to life imprisonment.
Der Richter verhängte eine lebenslange Freiheitsstrafe.
The judge imposed a life sentence.